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class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Төп меню</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Төп меню</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%90%D2%93%D1%8B%D0%BC%D0%B4%D0%B0%D2%93%D1%8B_%D0%B2%D0%B0%D2%A1%D0%B8%D2%93%D0%B0%D0%BB%D0%B0%D1%80" title="Ағымдағы ваҡиғалар тураһында белешмә мәғлүмәт"><span>Ағымдағы ваҡиғалар</span></a></li> </ul> </div> </div> <div id="p-articles" class="vector-menu mw-portlet mw-portlet-articles" > <div class="vector-menu-heading"> Мәҡәләләр </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-featured" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D2%BA%D0%B0%D0%B9%D0%BB%D0%B0%D0%BD%D2%93%D0%B0%D0%BD_%D0%BC%D3%99%D2%A1%D3%99%D0%BB%D3%99%D0%BB%D3%99%D1%80"><span>Һайланған</span></a></li><li id="n-good" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%AF%D2%A1%D1%88%D1%8B_%D0%BC%D3%99%D2%A1%D3%99%D0%BB%D3%99%D0%BB%D3%99%D1%80"><span>Яҡшы</span></a></li><li id="n-quality" class="mw-list-item"><a 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cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Википедия эсендә эҙләү" aria-label="Википедия эсендә эҙләү" autocapitalize="sentences" title="Википедия эсендә эҙләү [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Махсус:Эҙләү"> </div> <button class="cdx-button cdx-search-input__end-button">Эҙләргә</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Шәхси ҡоралдар"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Уҡыу көйләүҙәре"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Измените внешний вид страницы, размер, ширину и цвет шрифта." > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Уҡыу көйләүҙәре" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Уҡыу көйләүҙәре</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=ba" class=""><span>Иғәнә</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:%D0%98%D2%AB%D3%99%D0%BF_%D1%8F%D2%99%D1%8B%D1%83%D1%8B_%D1%8F%D2%BB%D0%B0%D1%83&returnto=%D2%A0%D1%83%D0%BB%D1%81%D0%B0+%28%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%29" title="Мотлаҡ булмаһа ла, иҫәп яҙмаһы төҙөргә һәм системала танылырға тәҡдим итәбеҙ" class=""><span>Теркәлеү</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:%D0%A2%D0%B0%D0%BD%D1%8B%D0%BB%D1%8B%D1%83&returnto=%D2%A0%D1%83%D0%BB%D1%81%D0%B0+%28%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%29" title="Бында теркәлеү үтергә була, әммә мотлаҡ түгел [o]" accesskey="o" class=""><span>Танылыу</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Күберәк мөмкинлектәр" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Шәхси ҡоралдар" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Шәхси ҡоралдар</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Ҡатнашыусы менюһы" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=ba"><span>Иғәнә</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:%D0%98%D2%AB%D3%99%D0%BF_%D1%8F%D2%99%D1%8B%D1%83%D1%8B_%D1%8F%D2%BB%D0%B0%D1%83&returnto=%D2%A0%D1%83%D0%BB%D1%81%D0%B0+%28%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%29" title="Мотлаҡ булмаһа ла, иҫәп яҙмаһы төҙөргә һәм системала танылырға тәҡдим итәбеҙ"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Теркәлеү</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:%D0%A2%D0%B0%D0%BD%D1%8B%D0%BB%D1%8B%D1%83&returnto=%D2%A0%D1%83%D0%BB%D1%81%D0%B0+%28%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%29" title="Бында теркәлеү үтергә була, әммә мотлаҡ түгел [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Танылыу</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Танылмаған мөхәррирҙәр өсөн бит <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A0%D3%99%D1%85%D0%B8%D0%BC_%D0%B8%D1%82%D0%B5%D0%B3%D0%B5%D2%99" aria-label="Мөхәррирләү тураһында ентеклерәк"><span>Күберәк белергә</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:%D3%A8%D0%BB%D3%A9%D1%88%D3%A9%D0%BC" title="Был IP-адрестан яһалған төҙәтеүҙәр [y]" accesskey="y"><span>Өлөш</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:%D3%98%D2%A3%D0%B3%D3%99%D0%BC%D3%99_%D0%B1%D0%B8%D1%82%D0%B5%D0%BC" title="IP-адресығыҙ өсөн фекер алышыу бите [n]" accesskey="n"><span>Фекер алышыу</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eйәшерергә\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"sitenotice\" lang=\"ba\" dir=\"ltr\"\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Сайт"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Йөкмәткеһе" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Йөкмәткеһе</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">ҡабырға панеленә күсерергә</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">йәшерергә</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Башы</div> </a> </li> <li id="toc-Тарихы" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Тарихы"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Тарихы</span> </div> </a> <ul id="toc-Тарихы-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Билдәләмә" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Билдәләмә"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Билдәләмә</span> </div> </a> <ul id="toc-Билдәләмә-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Иң_ябай_үҙсәнлектәре" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Иң_ябай_үҙсәнлектәре"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Иң ябай үҙсәнлектәре</span> </div> </a> <ul id="toc-Иң_ябай_үҙсәнлектәре-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Төп_төшөнсәләр" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Төп_төшөнсәләр"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Төп төшөнсәләр</span> </div> </a> <button aria-controls="toc-Төп_төшөнсәләр-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Төп төшөнсәләр бүлеген күрһәтергә/йәшерергә</span> </button> <ul id="toc-Төп_төшөнсәләр-sublist" class="vector-toc-list"> <li id="toc-Ҡулса_элементтары_төрҙәре" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ҡулса_элементтары_төрҙәре"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Ҡулса элементтары төрҙәре</span> </div> </a> <ul id="toc-Ҡулса_элементтары_төрҙәре-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Аҫҡулса" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Аҫҡулса"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Аҫҡулса</span> </div> </a> <ul id="toc-Аҫҡулса-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Идеалдар" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Идеалдар"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Идеалдар</span> </div> </a> <ul id="toc-Идеалдар-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Гомоморфизм" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Гомоморфизм"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Гомоморфизм</span> </div> </a> <ul id="toc-Гомоморфизм-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Факторҡулса" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Факторҡулса"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Факторҡулса</span> </div> </a> <ul id="toc-Факторҡулса-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ҡулсаларҙың_ҡайһы_бер_махсус_кластары" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ҡулсаларҙың_ҡайһы_бер_махсус_кластары"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ҡулсаларҙың ҡайһы бер махсус кластары</span> </div> </a> <ul id="toc-Ҡулсаларҙың_ҡайһы_бер_махсус_кластары-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Миҫалдар" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Миҫалдар"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Миҫалдар</span> </div> </a> <ul id="toc-Миҫалдар-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Конструкциялар" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Конструкциялар"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Конструкциялар</span> </div> </a> <button aria-controls="toc-Конструкциялар-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Конструкциялар бүлеген күрһәтергә/йәшерергә</span> </button> <ul id="toc-Конструкциялар-sublist" class="vector-toc-list"> <li id="toc-Тура_ҡабатландыҡ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Тура_ҡабатландыҡ"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Тура ҡабатландыҡ</span> </div> </a> <ul id="toc-Тура_ҡабатландыҡ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Эндоморфизмдар_ҡулсаһы" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Эндоморфизмдар_ҡулсаһы"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Эндоморфизмдар ҡулсаһы</span> </div> </a> <ul id="toc-Эндоморфизмдар_ҡулсаһы-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Бүлендектәр_яланы_һәм_бүлендектәр_ҡулсаһы" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Бүлендектәр_яланы_һәм_бүлендектәр_ҡулсаһы"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Бүлендектәр яланы һәм бүлендектәр ҡулсаһы</span> </div> </a> <ul id="toc-Бүлендектәр_яланы_һәм_бүлендектәр_ҡулсаһы-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Категориялы_һүрәтләү" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Категориялы_һүрәтләү"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Категориялы һүрәтләү</span> </div> </a> <ul id="toc-Категориялы_һүрәтләү-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ҡулсаларҙың_махсус_кластары" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ҡулсаларҙың_махсус_кластары"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Ҡулсаларҙың махсус кластары</span> </div> </a> <ul id="toc-Ҡулсаларҙың_махсус_кластары-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ҡулсалар_өҫтөндә_структуралар" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ҡулсалар_өҫтөндә_структуралар"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Ҡулсалар өҫтөндә структуралар</span> </div> </a> <ul id="toc-Ҡулсалар_өҫтөндә_структуралар-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Иҫкәрмәләр" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Иҫкәрмәләр"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Иҫкәрмәләр</span> </div> </a> <ul id="toc-Иҫкәрмәләр-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Әҙәбиәт" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Әҙәбиәт"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Әҙәбиәт</span> </div> </a> <ul id="toc-Әҙәбиәт-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Йөкмәткеһе" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Йөкмәткене күрһәтергә/йәшерергә" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Йөкмәткене күрһәтергә/йәшерергә</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ҡулса (математика)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Икенсе телдәге мәҡәләгә күсегеҙ. 66 телдә уҡырға була" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-66" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">66 тел</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%84%D9%82%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="حلقة (رياضيات) — арабский" lang="ar" hreflang="ar" data-title="حلقة (رياضيات)" data-language-autonym="العربية" data-language-local-name="арабский" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BB%D1%8C%D1%86%D0%BE_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Кальцо (алгебра) — белорусский" lang="be" hreflang="be" data-title="Кальцо (алгебра)" data-language-autonym="Беларуская" data-language-local-name="белорусский" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D1%8A%D1%81%D1%82%D0%B5%D0%BD_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Пръстен (алгебра) — болгарский" lang="bg" hreflang="bg" data-title="Пръстен (алгебра)" data-language-autonym="Български" data-language-local-name="болгарский" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Prsten_(matematika)" title="Prsten (matematika) — боснийский" lang="bs" hreflang="bs" data-title="Prsten (matematika)" data-language-autonym="Bosanski" data-language-local-name="боснийский" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Anell_(matem%C3%A0tiques)" title="Anell (matemàtiques) — каталанский" lang="ca" hreflang="ca" data-title="Anell (matemàtiques)" data-language-autonym="Català" data-language-local-name="каталанский" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Okruh_(algebra)" title="Okruh (algebra) — чешский" lang="cs" hreflang="cs" data-title="Okruh (algebra)" data-language-autonym="Čeština" data-language-local-name="чешский" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D0%BD%D0%BA%C4%83_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ункă (математика) — чувашский" lang="cv" hreflang="cv" data-title="Ункă (математика)" data-language-autonym="Чӑвашла" data-language-local-name="чувашский" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Modrwy_(mathemateg)" title="Modrwy (mathemateg) — валлийский" lang="cy" hreflang="cy" data-title="Modrwy (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="валлийский" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ring_(matematik)" title="Ring (matematik) — датский" lang="da" hreflang="da" data-title="Ring (matematik)" data-language-autonym="Dansk" data-language-local-name="датский" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ring_(Algebra)" title="Ring (Algebra) — немецкий" lang="de" hreflang="de" data-title="Ring (Algebra)" data-language-autonym="Deutsch" data-language-local-name="немецкий" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B1%CE%BA%CF%84%CF%8D%CE%BB%CE%B9%CE%BF%CF%82_(%CE%AC%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1)" title="Δακτύλιος (άλγεβρα) — греческий" lang="el" hreflang="el" data-title="Δακτύλιος (άλγεβρα)" data-language-autonym="Ελληνικά" data-language-local-name="греческий" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Ring_(mathematics)" title="Ring (mathematics) — английский" lang="en" hreflang="en" data-title="Ring (mathematics)" data-language-autonym="English" data-language-local-name="английский" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ringo_(algebro)" title="Ringo (algebro) — эсперанто" lang="eo" hreflang="eo" data-title="Ringo (algebro)" data-language-autonym="Esperanto" data-language-local-name="эсперанто" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Anillo_(matem%C3%A1tica)" title="Anillo (matemática) — испанский" lang="es" hreflang="es" data-title="Anillo (matemática)" data-language-autonym="Español" data-language-local-name="испанский" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ring_(algebra)" title="Ring (algebra) — эстонский" lang="et" hreflang="et" data-title="Ring (algebra)" data-language-autonym="Eesti" data-language-local-name="эстонский" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eraztun_(matematika)" title="Eraztun (matematika) — баскский" lang="eu" hreflang="eu" data-title="Eraztun (matematika)" data-language-autonym="Euskara" data-language-local-name="баскский" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D9%84%D9%82%D9%87_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="حلقه (ریاضیات) — персидский" lang="fa" hreflang="fa" data-title="حلقه (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="персидский" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Rengas_(matematiikka)" title="Rengas (matematiikka) — финский" lang="fi" hreflang="fi" data-title="Rengas (matematiikka)" data-language-autonym="Suomi" data-language-local-name="финский" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Anneau_(math%C3%A9matiques)" title="Anneau (mathématiques) — французский" lang="fr" hreflang="fr" data-title="Anneau (mathématiques)" data-language-autonym="Français" data-language-local-name="французский" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/F%C3%A1inne_(matamaitic)" title="Fáinne (matamaitic) — ирландский" lang="ga" hreflang="ga" data-title="Fáinne (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="ирландский" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Anel_(%C3%A1lxebra)" title="Anel (álxebra) — галисийский" lang="gl" hreflang="gl" data-title="Anel (álxebra)" data-language-autonym="Galego" data-language-local-name="галисийский" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%92_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="חוג (מבנה אלגברי) — иврит" lang="he" hreflang="he" data-title="חוג (מבנה אלגברי)" data-language-autonym="עברית" data-language-local-name="иврит" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Prsten_(matematika)" title="Prsten (matematika) — хорватский" lang="hr" hreflang="hr" data-title="Prsten (matematika)" data-language-autonym="Hrvatski" data-language-local-name="хорватский" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gy%C5%B1r%C5%B1_(matematika)" title="Gyűrű (matematika) — венгерский" lang="hu" hreflang="hu" data-title="Gyűrű (matematika)" data-language-autonym="Magyar" data-language-local-name="венгерский" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%95%D5%B2%D5%A1%D5%AF_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Օղակ (մաթեմատիկա) — армянский" lang="hy" hreflang="hy" data-title="Օղակ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="армянский" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Anello_(algebra)" title="Anello (algebra) — интерлингва" lang="ia" hreflang="ia" data-title="Anello (algebra)" data-language-autonym="Interlingua" data-language-local-name="интерлингва" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gelanggang_(matematika)" title="Gelanggang (matematika) — индонезийский" lang="id" hreflang="id" data-title="Gelanggang (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="индонезийский" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Anello_(algebra)" title="Anello (algebra) — итальянский" lang="it" hreflang="it" data-title="Anello (algebra)" data-language-autonym="Italiano" data-language-local-name="итальянский" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%92%B0_(%E6%95%B0%E5%AD%A6)" title="環 (数学) — японский" lang="ja" hreflang="ja" data-title="環 (数学)" data-language-autonym="日本語" data-language-local-name="японский" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A0%E1%83%92%E1%83%9D%E1%83%9A%E1%83%98_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="რგოლი (მათემატიკა) — грузинский" lang="ka" hreflang="ka" data-title="რგოლი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="грузинский" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B0%E0%B2%BF%E0%B2%82%E0%B2%97%E0%B3%8D" title="ರಿಂಗ್ — каннада" lang="kn" hreflang="kn" data-title="ರಿಂಗ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="каннада" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학) — корейский" lang="ko" hreflang="ko" data-title="환 (수학)" data-language-autonym="한국어" data-language-local-name="корейский" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Anellus" title="Anellus — латинский" lang="la" hreflang="la" data-title="Anellus" data-language-autonym="Latina" data-language-local-name="латинский" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Rank_(Algeber)" title="Rank (Algeber) — люксембургский" lang="lb" hreflang="lb" data-title="Rank (Algeber)" data-language-autonym="Lëtzebuergesch" data-language-local-name="люксембургский" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Anell_(matematega)" title="Anell (matematega) — Lombard" lang="lmo" hreflang="lmo" data-title="Anell (matematega)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%B2%E0%B4%AF%E0%B4%82_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="വലയം (ഗണിതം) — малаялам" lang="ml" hreflang="ml" data-title="വലയം (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="малаялам" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B0%E0%A4%BF%E0%A4%82%E0%A4%97" title="रिंग — маратхи" lang="mr" hreflang="mr" data-title="रिंग" data-language-autonym="मराठी" data-language-local-name="маратхи" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Gelanggang_(matematik)" title="Gelanggang (matematik) — малайский" lang="ms" hreflang="ms" data-title="Gelanggang (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="малайский" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ring_(wiskunde)" title="Ring (wiskunde) — нидерландский" lang="nl" hreflang="nl" data-title="Ring (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="нидерландский" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ring_i_matematikk" title="Ring i matematikk — нюнорск" lang="nn" hreflang="nn" data-title="Ring i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="нюнорск" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ring_(matematikk)" title="Ring (matematikk) — норвежский букмол" lang="nb" hreflang="nb" data-title="Ring (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="норвежский букмол" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Ringe_(matematike)" title="Ringe (matematike) — Novial" lang="nov" hreflang="nov" data-title="Ringe (matematike)" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pier%C5%9Bcie%C5%84_(matematyka)" title="Pierścień (matematyka) — польский" lang="pl" hreflang="pl" data-title="Pierścień (matematyka)" data-language-autonym="Polski" data-language-local-name="польский" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Anel" title="Anel — Piedmontese" lang="pms" hreflang="pms" data-title="Anel" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Anel_(matem%C3%A1tica)" title="Anel (matemática) — португальский" lang="pt" hreflang="pt" data-title="Anel (matemática)" data-language-autonym="Português" data-language-local-name="португальский" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Inel_(matematic%C4%83)" title="Inel (matematică) — румынский" lang="ro" hreflang="ro" data-title="Inel (matematică)" data-language-autonym="Română" data-language-local-name="румынский" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D1%8C%D1%86%D0%BE_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Кольцо (математика) — русский" lang="ru" hreflang="ru" data-title="Кольцо (математика)" data-language-autonym="Русский" data-language-local-name="русский" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Aneddu_(matim%C3%A0tica)" title="Aneddu (matimàtica) — сицилийский" lang="scn" hreflang="scn" data-title="Aneddu (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="сицилийский" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Algebarski_prsten" title="Algebarski prsten — сербскохорватский" lang="sh" hreflang="sh" data-title="Algebarski prsten" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="сербскохорватский" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Ring_(mathematics)" title="Ring (mathematics) — Simple English" lang="en-simple" hreflang="en-simple" data-title="Ring (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Okruh_(algebra)" title="Okruh (algebra) — словацкий" lang="sk" hreflang="sk" data-title="Okruh (algebra)" data-language-autonym="Slovenčina" data-language-local-name="словацкий" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kolobar_(algebra)" title="Kolobar (algebra) — словенский" lang="sl" hreflang="sl" data-title="Kolobar (algebra)" data-language-autonym="Slovenščina" data-language-local-name="словенский" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D0%B0%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D1%81%D1%82%D0%B5%D0%BD" title="Алгебарски прстен — сербский" lang="sr" hreflang="sr" data-title="Алгебарски прстен" data-language-autonym="Српски / srpski" data-language-local-name="сербский" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ring_(matematik)" title="Ring (matematik) — шведский" lang="sv" hreflang="sv" data-title="Ring (matematik)" data-language-autonym="Svenska" data-language-local-name="шведский" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%AF%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="வளையம் (கணிதம்) — тамильский" lang="ta" hreflang="ta" data-title="வளையம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="тамильский" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B4%E0%B8%87_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="ริง (คณิตศาสตร์) — тайский" lang="th" hreflang="th" data-title="ริง (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="тайский" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Halka" title="Halka — турецкий" lang="tr" hreflang="tr" data-title="Halka" data-language-autonym="Türkçe" data-language-local-name="турецкий" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D1%96%D0%BB%D1%8C%D1%86%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Кільце (алгебра) — украинский" lang="uk" hreflang="uk" data-title="Кільце (алгебра)" data-language-autonym="Українська" data-language-local-name="украинский" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D9%84%D9%82%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="حلقہ (ریاضی) — урду" lang="ur" hreflang="ur" data-title="حلقہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="урду" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/V%C3%A0nh" title="Vành — вьетнамский" lang="vi" hreflang="vi" data-title="Vành" data-language-autonym="Tiếng Việt" data-language-local-name="вьетнамский" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Rienk_(algebra)" title="Rienk (algebra) — West Flemish" lang="vls" hreflang="vls" data-title="Rienk (algebra)" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%8E%AF%EF%BC%88%E4%BB%A3%E6%95%B0%EF%BC%89" title="环（代数） — у" lang="wuu" hreflang="wuu" data-title="环（代数）" data-language-autonym="吴语" data-language-local-name="у" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%8E%AF_(%E4%BB%A3%E6%95%B0)" title="环 (代数) — китайский" lang="zh" hreflang="zh" data-title="环 (代数)" data-language-autonym="中文" data-language-local-name="китайский" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%92%B0_(%E4%BB%A3%E6%95%B8)" title="環 (代數) — Literary Chinese" lang="lzh" hreflang="lzh" data-title="環 (代數)" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Kh%C3%B4an" title="Khôan — миньнань" lang="nan" hreflang="nan" data-title="Khôan" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="миньнань" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%92%B0_(%E4%BB%A3%E6%95%B8)" title="環 (代數) — кантонский" lang="yue" hreflang="yue" data-title="環 (代數)" data-language-autonym="粵語" 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href="/w/index.php?title=%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:UrlShortener&url=https%3A%2F%2Fba.wikipedia.org%2Fwiki%2F%25D2%25A0%25D1%2583%25D0%25BB%25D1%2581%25D0%25B0_%28%25D0%25BC%25D0%25B0%25D1%2582%25D0%25B5%25D0%25BC%25D0%25B0%25D1%2582%25D0%25B8%25D0%25BA%25D0%25B0%29"><span>Ҡыҫҡа URL алыу</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:QrCode&url=https%3A%2F%2Fba.wikipedia.org%2Fwiki%2F%25D2%25A0%25D1%2583%25D0%25BB%25D1%2581%25D0%25B0_%28%25D0%25BC%25D0%25B0%25D1%2582%25D0%25B5%25D0%25BC%25D0%25B0%25D1%2582%25D0%25B8%25D0%25BA%25D0%25B0%29"><span>QR код алыу</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Баҫтырыу/сығарыу </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a 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class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q161172" title="Мәғлүмәт һаҡлағысына бәйле элементҡа һылтанма [g]" accesskey="g"><span>Викимәғлүмәт элементы</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Ҡоралдар"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Уҡыу көйләүҙәре"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Уҡыу көйләүҙәре</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">ҡабырға панеленә күсерергә</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">йәшерергә</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Википедия — ирекле энциклопедия мәғлүмәте</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ba" dir="ltr"><div class="dablink noprint">Был терминдың башҡа мәғәнәләре лә бар, ҡарағыҙ: <a href="/w/index.php?title=%D0%9A%D0%BE%D0%BB%D1%8C%D1%86%D0%BE&action=edit&redlink=1" class="new" title="Кольцо (был бит юҡ)">Кольцо</a>.</div> <p><a href="/w/index.php?title=%D2%A0%D0%B0%D0%BB%D1%8B%D0%BF:About&action=edit&redlink=1" class="new" title="Ҡалып:About (был бит юҡ)">Ҡалып:About</a> <b>Ҡулса</b> (шулай уҡ <i>ассоциатив ҡулса</i>) <a href="/wiki/%D0%94%D3%A9%D0%B9%D3%A9%D0%BC_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Дөйөм алгебра">дөйөм алгебрала</a> — үҙсәнлектәре буйынса <a href="/wiki/%D2%BA%D0%B0%D0%BD" title="Һан">һандар</a> өҫтөндәге ярашлы операцияларға оҡшаш әйләндерелмәле <a href="/wiki/%D2%A0%D1%83%D1%88%D1%8B%D1%83" title="Ҡушыу">ҡушыу</a> операцияһы һәм <a href="/wiki/%D2%A0%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D1%83" title="Ҡабатлау">ҡабатлау</a> операцияһы бирелгән <a href="/w/index.php?title=%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA%D1%8F_%D1%81%D1%82%D1%80%D1%83%D0%BA%D1%82%D1%83%D1%80%D0%B0&action=edit&redlink=1" class="new" title="Алгебраикя структура (был бит юҡ)">алгебраикя структура</a>. Ҡулсаларҙың иң ябай миҫалдары булып һандар йыйылмалары (<a href="/wiki/%D0%91%D3%A9%D1%82%D3%A9%D0%BD_%D2%BB%D0%B0%D0%BD" title="Бөтөн һан">бөтөн</a>, <a href="/w/index.php?title=%D0%AB%D1%81%D1%8B%D0%BD_%D2%BB%D0%B0%D0%BD%D0%B4%D0%B0%D1%80&action=edit&redlink=1" class="new" title="Ысын һандар (был бит юҡ)">ысын</a>, <a href="/w/index.php?title=%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%D1%8B_%D2%BB%D0%B0%D0%BD%D0%B4%D0%B0%D1%80&action=edit&redlink=1" class="new" title="Комплекслы һандар (был бит юҡ)">комплекслы</a>), бирелгән күмәклектә билдәләнгән һанлы <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика)">функциялар</a> йыйылмалары торалар. Бөтә осраҡтарҙа ла һандар йыйылмаһына оҡшаш күмәклектәр бар, уларҙың <a href="/w/index.php?title=%D0%9A%D2%AF%D0%BC%D3%99%D0%BA%D0%BB%D0%B5%D0%BA_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D1%8B&action=edit&redlink=1" class="new" title="Күмәклек элементы (был бит юҡ)">элементтарын</a> ҡушырға һәм ҡабатларға мөмкин, шуның менән бергә был операциялар үҙҙәрен тәбиғи рәүештә тоталар<sup id="cite_ref-Винберг—2011——17—19_1-0" class="reference"><a href="#cite_note-Винберг—2011——17—19-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. </p><p>Ҡушыу һәм ҡабатлау операцияларының дөйөм үҙсәнлектәрен, уларҙың операциялар башҡарылған элементтарҙың тәбиғәтенә ҡағылышһыҙ үҙ-ара эске бәйләнешен өйрәнеү өсөн ҡулса төшөнсәһе индерелә лә инде<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. </p><p>Ҡулсалар <a href="/wiki/%D2%A0%D1%83%D0%BB%D1%81%D0%B0%D0%BB%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Ҡулсалар теорияһы">ҡулсалар теорияһының</a> — дөйөм алгебраның ҙур бүлегенең төп өйрәнеү объекты булып торалар, унда <a href="/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Алгебраик геометрия">алгебраик геометрияла</a>, <a href="/w/index.php?title=%D2%BA%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D2%99%D1%8B%D2%A3_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Һандарҙың алгебраик теорияһы (был бит юҡ)">һандарҙың алгебраик теорияһында</a>, <a href="/w/index.php?title=%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_K-%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Алгебраик K-теория (был бит юҡ)">алгебраик <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>-теорияла</a>, <a href="/w/index.php?title=%D0%98%D0%BD%D0%B2%D0%B0%D1%80%D0%B8%D0%B0%D0%BD%D1%82%D1%82%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Инварианттар теорияһы (был бит юҡ)">инварианттар теорияһында</a> киң ҡулланыу тапҡан инструменталь саралар эшләнгән. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Тарихы"><span id=".D0.A2.D0.B0.D1.80.D0.B8.D1.85.D1.8B"></span>Тарихы</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=1" title="Тарихы бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=1" title="Тарихы бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Алгебраның фән булараҡ йылдам үҫеше XIX быуатта башлана. Һандар теорияһының төп бурыстарының береһе булып 1860—1870-се йылдарҙа <a href="/w/index.php?title=%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D2%BB%D0%B0%D0%BD&action=edit&redlink=1" class="new" title="Алгебраик һан (был бит юҡ)">алгебраик һандарҙың</a> дөйөм яландарында <a href="/w/index.php?title=%D0%91%D2%AF%D0%BB%D0%B5%D0%BD%D0%B5%D2%AF%D1%81%D3%99%D0%BD%D0%BB%D0%B5%D0%BA&action=edit&redlink=1" class="new" title="Бүленеүсәнлек (был бит юҡ)">бүленеүсәнлек теорияһын</a> төҙөү тора. Был мәсьәләнең хәл ителеше <a href="/w/index.php?title=%D0%94%D0%B5%D0%B4%D0%B5%D0%BA%D0%B8%D0%BD%D0%B4,_%D0%AE%D0%BB%D0%B8%D1%83%D1%81_%D0%92%D0%B8%D0%BB%D1%8C%D0%B3%D0%B5%D0%BB%D1%8C%D0%BC_%D0%A0%D0%B8%D1%85%D0%B0%D1%80%D0%B4&action=edit&redlink=1" class="new" title="Дедекинд, Юлиус Вильгельм Рихард (был бит юҡ)">Рихард Дедекинд</a> тарафынан баҫтырылып сығарыла («X Дополнение к лекциям по теории чисел Дирихле», 1871 йыл). Был хеҙмәттә беренсе тапҡыр һанлы яландың <a href="/w/index.php?title=%D0%91%D3%A9%D1%82%D3%A9%D0%BD_%D2%BB%D0%B0%D0%BD%D0%B4%D0%B0%D1%80_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Бөтөн һандар ҡулсаһы (был бит юҡ)">бөтөн һандар ҡулсаһы</a> төшөнсәһе ҡарала, был контекста модуль һәм <a href="/w/index.php?title=%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&action=edit&redlink=1" class="new" title="Идеал (алгебра) (был бит юҡ)">идеал</a> төшөнсәләренә билдәләмә бирелә<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>. <a href="/w/index.php?title=%D2%A0%D0%B0%D0%BB%D1%8B%D0%BF:%D0%A0%D0%B0%D0%B7%D0%B4%D0%B5%D0%BB_%D0%BD%D0%B5_%D0%B7%D0%B0%D0%B2%D0%B5%D1%80%D1%88%D1%91%D0%BD&action=edit&redlink=1" class="new" title="Ҡалып:Раздел не завершён (был бит юҡ)">Ҡалып:Раздел не завершён</a> </p> <div class="mw-heading mw-heading2"><h2 id="Билдәләмә"><span id=".D0.91.D0.B8.D0.BB.D0.B4.D3.99.D0.BB.D3.99.D0.BC.D3.99"></span>Билдәләмә</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=2" title="Билдәләмә бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=2" title="Билдәләмә бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ҡулса — теләһә ниндәй <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155a175e62376943eefca56b9596ea133f93fe1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.907ex; height:2.509ex;" alt="{\displaystyle a,b,c\in R}"></span> өсөн түбәндәге үҙсәнлектәр үтәлгән, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span> (<b>ҡушыу</b> һәм <b>ҡабатлау</b> тип аталған) ике <a href="/w/index.php?title=%D0%91%D0%B8%D0%BD%D0%B0%D1%80_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Бинар операция (был бит юҡ)">бинар операция</a> бирелгән <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> <a href="/wiki/%D0%9A%D2%AF%D0%BC%D3%99%D0%BA%D0%BB%D0%B5%D0%BA" title="Күмәклек">күмәклеге</a> ул: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=b+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684f43b5094501674e8314be5e24a80ee64682e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}"></span> — ҡушыуҙың <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" class="mw-redirect" title="Коммутатив операция">коммутативлығы</a>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(b+c)=(a+b)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(b+c)=(a+b)+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44038eb287a7d11c82ecf1642362bff63a012b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle a+(b+c)=(a+b)+c}"></span> — ҡушыуҙың <a href="/w/index.php?title=%D0%90%D1%81%D1%81%D0%BE%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Ассоциатив операция (был бит юҡ)">ассоциативлығы</a>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists 0\in R\ \left(a+0=0+a=a\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mn>0</mn> <mo>∈</mo> <mi>R</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists 0\in R\ \left(a+0=0+a=a\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734e44867d17c2d8488ea2d8306001b562de1c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.729ex; height:2.843ex;" alt="{\displaystyle \exists 0\in R\ \left(a+0=0+a=a\right)}"></span> — ҡушыуға ҡарата <a href="/w/index.php?title=%D0%9D%D0%B5%D0%B9%D1%82%D1%80%D0%B0%D0%BB%D1%8C_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82&action=edit&redlink=1" class="new" title="Нейтраль элемент (был бит юҡ)">нейтраль элементтың</a> булыуы;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a\in R\;\exists b\in R\left(a+b=b+a=0\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>R</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>b</mi> <mo>∈</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a\in R\;\exists b\in R\left(a+b=b+a=0\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d07888c86657ac3e0c139bc064058322848c30b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.358ex; height:2.843ex;" alt="{\displaystyle \forall a\in R\;\exists b\in R\left(a+b=b+a=0\right)}"></span> — ҡушыуға ҡарата ҡапма-ҡаршы элементтың булыуы;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{matrix}a\times (b+c)=(a\times b)+(a\times c)\\(b+c)\times a=(b\times a)+(c\times a)\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>×</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>×</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>×</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{matrix}a\times (b+c)=(a\times b)+(a\times c)\\(b+c)\times a=(b\times a)+(c\times a)\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae8e1321f5d909037d2b1964f7b1129e8e8f48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.921ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{matrix}a\times (b+c)=(a\times b)+(a\times c)\\(b+c)\times a=(b\times a)+(c\times a)\end{matrix}}\right.}"></span> — <a href="/wiki/%D0%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D2%A1" title="Дистрибутивлыҡ">дистрибутивлыҡ</a>.</li></ol> <p>Икенсе төрлө әйткәндә, ҡулса — ҡушыуға <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> ҡарата <a href="/wiki/%D0%90%D0%B1%D0%B5%D0%BB%D1%8C_%D1%82%D3%A9%D1%80%D0%BA%D3%A9%D0%BC%D3%A9" title="Абель төркөмө">Абель төркөмө</a> булған һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span>-ҙың <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span>-ға ҡарата ике яҡлы <a href="/wiki/%D0%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D2%A1" title="Дистрибутивлыҡ">дистрибутивлығына</a> эйә булған <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(R,+,\times \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>×</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(R,+,\times \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e948dfce36abbb75b3dbf92e29e0da64b1f41fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.257ex; height:2.843ex;" alt="{\displaystyle \left(R,+,\times \right)}"></span> <a href="/w/index.php?title=%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_(%D1%83%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B0%D0%BB%D1%8C_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&action=edit&redlink=1" class="new" title="Алгебра (универсаль алгебра) (был бит юҡ)">универсаль алгебра</a> ул, ғәҙәттә ҡулса төшөнсәһе аҫтында ҡабатлауға ҡарата ассоциатив ҡулсаларҙы күҙ уңында тоталар, йәғни уларҙа мультипликатив төркөм ярым төркөм була. </p><p>Ҡулсалар түбәндәге өҫтәлмә үҙсәнлектәргә эйә булырға мөмкиндәр: </p> <ul><li><a href="/w/index.php?title=%D0%9D%D0%B5%D0%B9%D1%82%D1%80%D0%B0%D0%BB%D1%8C_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82&action=edit&redlink=1" class="new" title="Нейтраль элемент (был бит юҡ)">берәмектең</a> булыуы: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists e\in R\;\forall a\in R\left(a\times e=e\times a=a\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>e</mi> <mo>∈</mo> <mi>R</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>×</mo> <mi>e</mi> <mo>=</mo> <mi>e</mi> <mo>×</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists e\in R\;\forall a\in R\left(a\times e=e\times a=a\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4ccb58d40bd632fbee02bc964b213addce1d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.683ex; height:2.843ex;" alt="{\displaystyle \exists e\in R\;\forall a\in R\left(a\times e=e\times a=a\right)}"></span> (<i>берәмеге булған ҡулса</i>), берәмек ғәҙәттә 1 тип тамғалана;</li> <li>ҡабатлауҙың <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D2%A1" title="Коммутативлыҡ">коммутативлығы</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a,b\in R\left(a\times b=b\times a\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>×</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,b\in R\left(a\times b=b\times a\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00dc543bac1175651d2b7375033d5b65e1523a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.589ex; height:2.843ex;" alt="{\displaystyle \forall a,b\in R\left(a\times b=b\times a\right)}"></span> (<i><a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2_%D2%A1%D1%83%D0%BB%D1%81%D0%B0" title="Коммутатив ҡулса">коммутатив ҡулса</a></i>);</li></ul> <p>Ҡайһы берҙә ҡулса төшөнсәһе аҫтында тик <a href="/w/index.php?title=%D0%91%D0%B5%D1%80%D3%99%D0%BC%D0%B5%D0%BA_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&action=edit&redlink=1" class="new" title="Берәмек (алгебра) (был бит юҡ)">берәмеге булған</a> ҡулсаларҙы ғына аңлайҙар<sup id="cite_ref-Атья,_Макдональд—1972——9_4-0" class="reference"><a href="#cite_note-Атья,_Макдональд—1972——9-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> (йәғни <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(R,\times \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mo>,</mo> <mo>×</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(R,\times \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6bc22ff882032ccf23a297dec87b7d14f77dfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.415ex; height:2.843ex;" alt="{\displaystyle \left(R,\times \right)}"></span> <a href="/w/index.php?title=%D0%9C%D0%BE%D0%BD%D0%BE%D0%B8%D0%B4&action=edit&redlink=1" class="new" title="Моноид (был бит юҡ)">моноид</a> булыуын талап итәләр), ләкин берәмеге булмаған ҡулсалар ҙа өйрәнеләләр (мәҫәлән, йоп һандар ҡулсаһы берәмеге булмаған коммутатив ассоциатив ҡулса<sup id="cite_ref-Винберг—2011——18—19_5-0" class="reference"><a href="#cite_note-Винберг—2011——18—19-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>). </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span> символы урынына йыш ҡына <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"></span> символын ҡулланалар, йәки уны бөтөнләй төшөрөп ҡалдыралар. </p> <div class="mw-heading mw-heading2"><h2 id="Иң_ябай_үҙсәнлектәре"><span id=".D0.98.D2.A3_.D1.8F.D0.B1.D0.B0.D0.B9_.D2.AF.D2.99.D1.81.D3.99.D0.BD.D0.BB.D0.B5.D0.BA.D1.82.D3.99.D1.80.D0.B5"></span>Иң ябай үҙсәнлектәре</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=3" title="Иң ябай үҙсәнлектәре бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=3" title="Иң ябай үҙсәнлектәре бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ҡулсаның билдәләмәһенән туранан-тура түбәндәге үҙсәнлектәрҙе сығарырға мөмкин: </p> <ul><li>ҡулсала ҡушыуға ҡарата нейтраль элемент берҙән бер;</li> <li>ҡулсаның теләһә ниндәй элементы өсөн ҡушыуға ҡарата уға <a href="/w/index.php?title=%D0%9A%D0%B8%D1%80%D0%B5_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82&action=edit&redlink=1" class="new" title="Кире элемент (был бит юҡ)">кире</a> элемент берҙән бер;</li> <li>ҡабатлауға ҡарата нейтраль элемент, әгәр ул булһа, ул берҙән бер;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot 0=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot 0=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540d7a7b6f63e28d8cc6fa99854ac2f598f40570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.979ex; height:2.509ex;" alt="{\displaystyle a\cdot 0=0,}"></span> йәғни 0 — ҡабатлау буйынса йотоусы элемент;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-b)=(-1)\cdot b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-b)=(-1)\cdot b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029fbbec3b78b6b1fe75c32b6f20178f4745c393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.817ex; height:2.843ex;" alt="{\displaystyle (-b)=(-1)\cdot b,}"></span> бында <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48827b58299886483952c6fcf3f8f55948ab2fc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.615ex; height:2.843ex;" alt="{\displaystyle (-b)}"></span> — ҡушыу буйынса <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>-ға кире элемент;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-a)\cdot b=(-ab);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-a)\cdot b=(-ab);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764b273a2f809e14a4a1d76b23d7b3a460d13481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.114ex; height:2.843ex;" alt="{\displaystyle (-a)\cdot b=(-ab);}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-a)\cdot (-b)=(ab).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-a)\cdot (-b)=(ab).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5094c4f43b1b09a19ff7f756b1cdce46c5007a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.923ex; height:2.843ex;" alt="{\displaystyle (-a)\cdot (-b)=(ab).}"></span><sup id="cite_ref-Курош—1968——273—275_6-0" class="reference"><a href="#cite_note-Курош—1968——273—275-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Винберг—2011——18—19_5-1" class="reference"><a href="#cite_note-Винберг—2011——18—19-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Төп_төшөнсәләр"><span id=".D0.A2.D3.A9.D0.BF_.D1.82.D3.A9.D1.88.D3.A9.D0.BD.D1.81.D3.99.D0.BB.D3.99.D1.80"></span>Төп төшөнсәләр</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=4" title="Төп төшөнсәләр бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=4" title="Төп төшөнсәләр бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ҡулса_элементтары_төрҙәре"><span id=".D2.A0.D1.83.D0.BB.D1.81.D0.B0_.D1.8D.D0.BB.D0.B5.D0.BC.D0.B5.D0.BD.D1.82.D1.82.D0.B0.D1.80.D1.8B_.D1.82.D3.A9.D1.80.D2.99.D3.99.D1.80.D0.B5"></span>Ҡулса элементтары төрҙәре</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=5" title="Ҡулса элементтары төрҙәре бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=5" title="Ҡулса элементтары төрҙәре бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ҡулсала нулдән айырмалы элементтар бар икән, ти. Ул саҡта һул <a href="/w/index.php?title=%D0%9D%D1%83%D0%BB%D0%B4%D0%B5%D2%A3_%D0%B1%D2%AF%D0%BB%D0%B5%D2%AF%D1%81%D0%B5%D2%BB%D0%B5&action=edit&redlink=1" class="new" title="Нулдең бүлеүсеһе (был бит юҡ)">нулдең бүлеүсеһе</a> — ул <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының нулдән айырмалы шундай <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> элементы, уның өсөн <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a59b821cbfde052e05d53bf5f013cbb97a6bbd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.488ex; height:2.176ex;" alt="{\displaystyle ab=0}"></span> булған, нулдән айырмалы <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> элементы бар. Оҡшаш рәүештә нулдең уң бүлеүсеһенә билдәләмә бирелә. Коммутатив ҡулсаларҙа был төшөнсәләр тап киләләр. Миҫал: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e120a3bd60fc89b495dd7ef6039465b7e6a703b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,1)}"></span> интервалында өҙлөкһөҙ функциялар ҡулсаһын ҡарайыҡ. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\max(0,x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\max(0,x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3748c43adcce6520cc09f8e5259f6b9db371cbe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.824ex; height:2.843ex;" alt="{\displaystyle f(x)=\max(0,x),}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\max(0,-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=\max(0,-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6a181036cf3e4166c9b2d3bac37c9de015adb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.823ex; height:2.843ex;" alt="{\displaystyle g(x)=\max(0,-x)}"></span> булһын, ул саҡта <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\neq 0,g\neq 0,fg=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mi>g</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mi>f</mi> <mi>g</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\neq 0,g\neq 0,fg=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4329f6b3d84c05b8c899a8e01bee70fb0e6a2070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.287ex; height:2.676ex;" alt="{\displaystyle f\neq 0,g\neq 0,fg=0,}"></span> йәғни <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle f,g}"></span> нулдең бүлеүселәре булып торалар. Бында <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88725631c1c1c441d1b0db0aed0e22246f162b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.54ex; height:2.676ex;" alt="{\displaystyle f\neq 0}"></span> шарты <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> нулдән айырмалы функция тигәнде аңлата, ләкин <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> функцияһы бер ҡасан да <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> ҡиммәте ҡабул итмәй тигәнде аңлатмай<sup id="cite_ref-Ван_дер_Варден—1975——51—53_7-0" class="reference"><a href="#cite_note-Ван_дер_Варден—1975——51—53-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p><a href="/w/index.php?title=%D0%9D%D0%B8%D0%BB%D1%8C%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%82%D0%BB%D1%8B_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82&action=edit&redlink=1" class="new" title="Нильпотентлы элемент (был бит юҡ)">Нильпотентлы элемент</a> — ул шундай <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> элементы, ниндәйҙер <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"></span> өсөн <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1b03312f135b5289da188c9a23006fb401e574" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.709ex; height:2.343ex;" alt="{\displaystyle a^{n}=0}"></span>. Миҫал: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502b7bdecb108de51751199de7746733b55e886a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.299ex; height:3.343ex;" alt="{\displaystyle {\bigl (}{\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}{\bigr )}}"></span> <a href="/w/index.php?title=%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&redlink=1" class="new" title="Матрица (математика) (был бит юҡ)">матрицаһы</a>. Нильпотентлы элемент һәр ваҡыт <a href="/w/index.php?title=%D0%9D%D1%83%D0%BB%D0%B4%D0%B5%D2%A3_%D0%B1%D2%AF%D0%BB%D0%B5%D2%AF%D1%81%D0%B5%D2%BB%D0%B5&action=edit&redlink=1" class="new" title="Нулдең бүлеүсеһе (был бит юҡ)">нулдең бүлеүсеһе</a> була (тик әгәр ҡулса бер нулдән генә томаһа), киреһе дөйөм осраҡта дөрөҫ түгел<sup id="cite_ref-Атья,_Макдональд—1972——11_8-0" class="reference"><a href="#cite_note-Атья,_Макдональд—1972——11-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>. </p><p><a href="/w/index.php?title=%D0%98%D0%B4%D0%B5%D0%BC%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%82%D0%BB%D1%8B_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82&action=edit&redlink=1" class="new" title="Идемпотентлы элемент (был бит юҡ)">Идемпотентлы элемент</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> — ул шундай элемент, бында <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\cdot e=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>⋅</mo> <mi>e</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cdot e=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b43fb6c2e14c5d3c2caeeb797df1cc2ff02a14b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.028ex; height:1.676ex;" alt="{\displaystyle e\cdot e=e}"></span> Мәҫәлән, теләһә ниндәй <a href="/w/index.php?title=%D0%9F%D1%80%D0%BE%D0%B5%D0%BA%D1%86%D0%B8%D1%8F%D0%BB%D0%B0%D1%83_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80%D1%8B&action=edit&redlink=1" class="new" title="Проекциялау операторы (был бит юҡ)">проекциялау операторы</a> идемпотентлы, айырып әйткәндә, түбәндәге: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b96bde53b6e8f01a9fac1ceae475e746410ff72d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.812ex; height:2.176ex;" alt="{\displaystyle 2\times 2.}"></span> матрицалар ҡулсаһында <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/497bead416edf9081c0dd6904d9fe7954f2d15aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.299ex; height:3.343ex;" alt="{\displaystyle {\bigl (}{\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}{\bigr )}}"></span><sup id="cite_ref-Ван_дер_Варден—1975——359_9-0" class="reference"><a href="#cite_note-Ван_дер_Варден—1975——359-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> — берәмеге булған <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының ирекле элементы булһа, ул саҡта <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>-ға һул кире элемент тип шундай <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{l}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{l}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55741e19a373c29ac7bac3c1dd32e8cb645d1c98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.563ex; height:3.343ex;" alt="{\displaystyle a_{l}^{-1}}"></span> элементы атала, бында <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{l}^{-1}a=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mi>a</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{l}^{-1}a=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72bd22a1ecf23a380d1c056d58d26fd49dc93aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.7ex; height:3.343ex;" alt="{\displaystyle a_{l}^{-1}a=1.}"></span> Уң кире элементҡа оҡшаш рәүештә билдәләмә бирелә. Әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> элементының уң элементы ла һәм һул элементы ла булһа, ул саҡта улар тап киләләр, һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> элементының берҙән бер аныҡланған һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5709c8d86f7fec8fb86069bf5d15a9eabe564e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.676ex;" alt="{\displaystyle a^{-1}}"></span> тип тамғаланған кире элементы бар тип әйтәләр. Элемент үҙе әйләндермәле элемент тип атала.<sup id="cite_ref-Ван_дер_Варден—1975——51—53_7-1" class="reference"><a href="#cite_note-Ван_дер_Варден—1975——51—53-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Аҫҡулса"><span id=".D0.90.D2.AB.D2.A1.D1.83.D0.BB.D1.81.D0.B0"></span>Аҫҡулса</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=6" title="Аҫҡулса бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=6" title="Аҫҡулса бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="dablink noprint">Төп мәҡәлә: <b><a href="/w/index.php?title=%D0%9F%D0%BE%D0%B4%D0%BA%D0%BE%D0%BB%D1%8C%D1%86%D0%BE&action=edit&redlink=1" class="new" title="Подкольцо (был бит юҡ)">Подкольцо</a></b> </div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700185d360e1c9ac519f2e60397267565178b579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset R}"></span> аҫкүмәклеге, әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-ҙа бирелгән операцияларға ҡарата үҙе ҡулса булһа, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-ҙың <b>аҫҡулсаһы</b>тип атала. Был осраҡта <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> ҡулсаһының киңәйеүе тип әйтәләр. <sup id="cite_ref-Винберг—2011——407_10-0" class="reference"><a href="#cite_note-Винберг—2011——407-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Икенсе төрлө әйткәндә, буш булмаған <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700185d360e1c9ac519f2e60397267565178b579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset R}"></span> аҫкүмәклеге аҫҡулса була, әгәр </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының аддитив аҫтөркөмө булһа, йәғни теләһә ниндәй <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd3c2bd17578ba23f55a299668cd7accc5c7a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.103ex; height:2.509ex;" alt="{\displaystyle x,y\in A}"></span> өсөн: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y,-x\in A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y,-x\in A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e685e167967d9083e80a67aaad06bdc6e85d852" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.728ex; height:2.509ex;" alt="{\displaystyle x+y,-x\in A,}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> ҡабатлауға ҡарата йомоҡ, йәғни теләһә ниндәй <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in A:xy\in A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mo>:</mo> <mi>x</mi> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in A:xy\in A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be8355d8a04d87c647e7406659f187f3868a37a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.756ex; height:2.509ex;" alt="{\displaystyle x,y\in A:xy\in A.}"></span></li></ul> <p><br /> Билдәләмә буйынса, аҫҡулса <a href="/w/index.php?title=%D0%91%D1%83%D1%88_%D0%B1%D1%83%D0%BB%D0%BC%D0%B0%D2%93%D0%B0%D0%BD_%D0%BA%D2%AF%D0%BC%D3%99%D0%BA%D0%BB%D0%B5%D0%BA&action=edit&redlink=1" class="new" title="Буш булмаған күмәклек (был бит юҡ)">буш түгел</a>, сөнки <a href="/w/index.php?title=%D0%9D%D1%83%D0%BB%D1%8C_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82&action=edit&redlink=1" class="new" title="Нуль элемент (был бит юҡ)">нуль элементы</a> бар. Ҡулсаның нуле һәм берәмеге уның теләһә ниндәй аҫҡулсаһының нуле һәм берәмеге була<sup id="cite_ref-Куликов—1979——110—111_11-0" class="reference"><a href="#cite_note-Куликов—1979——110—111-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>. </p><p>Аҫҡулса коммутативлыҡ үҙсәнлеген һаҡлап ҡала<sup id="cite_ref-Винберг—2011——21_12-0" class="reference"><a href="#cite_note-Винберг—2011——21-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>. </p><p>Теләһә ниндәй аҫҡулсалар күмәклеге <a href="/w/index.php?title=%D0%9A%D2%AF%D0%BC%D3%99%D0%BA%D0%BB%D0%B5%D0%BA%D1%82%D3%99%D1%80_%D0%BA%D0%B8%D2%AB%D0%B5%D0%BB%D0%B5%D1%88%D0%B5&action=edit&redlink=1" class="new" title="Күмәклектәр киҫелеше (был бит юҡ)">киҫелеше</a> аҫҡулса була. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subset R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊂</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subset R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9a0d58ce4606a6c044315305acdd78e5c12cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.638ex; height:2.176ex;" alt="{\displaystyle E\subset R}"></span> аҫкүмәклеге ингән иң бәләкәй аҫҡулса <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> барлыҡҡа килтергән аҫҡулса тип атала, ә <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһы өсөн барлыҡҡа килтереүсе система тип атала. Ундай аҫҡулса һәр ваҡыт бар, сөнки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> кергән бөтә аҫҡулсаларҙың киҫелеше был билдәләмәне ҡәнәғәтләндерә.<sup id="cite_ref-Куликов—1979——110—111_11-1" class="reference"><a href="#cite_note-Куликов—1979——110—111-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Берәмеге булған <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының, уның берәмеге барлыҡҡа килтергән аҫҡулсаһы, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының иң бәләкәй йәки төп аҫҡулсаһы тип атала. Бындай аҫҡулса <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының теләһә ниндәй аҫҡулсаһына инә<sup id="cite_ref-Куликов—1979——437_13-0" class="reference"><a href="#cite_note-Куликов—1979——437-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Идеалдар"><span id=".D0.98.D0.B4.D0.B5.D0.B0.D0.BB.D0.B4.D0.B0.D1.80"></span>Идеалдар</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=7" title="Идеалдар бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=7" title="Идеалдар бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="dablink noprint">Төп мәҡәлә: <b><a href="/w/index.php?title=%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&action=edit&redlink=1" class="new" title="Идеал (алгебра) (был бит юҡ)">Идеал (алгебра)</a></b> </div> <p>Ҡулса идеалының билдәләмәһе һәм роле <a href="/wiki/%D0%A2%D3%A9%D1%80%D0%BA%D3%A9%D0%BC_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Төркөм (математика)">төркөм</a> теорияһындағы <a href="/w/index.php?title=%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C_%D0%B0%D2%AB%D1%82%D3%A9%D1%80%D0%BA%D3%A9%D0%BC&action=edit&redlink=1" class="new" title="Нормаль аҫтөркөм (был бит юҡ)">нормаль аҫтөркөм</a> билдәләмәһе һәм роле менән оҡшаш <sup id="cite_ref-Ван_дер_Варден—1975——64_14-0" class="reference"><a href="#cite_note-Ван_дер_Варден—1975——64-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> буш булмаған аҫкүмәклеге һул идеал тип атала, әгәр: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> ҡулсаның аддитив <a href="/w/index.php?title=%D0%90%D2%AB%D1%82%D3%A9%D1%80%D0%BA%D3%A9%D0%BC&action=edit&redlink=1" class="new" title="Аҫтөркөм (был бит юҡ)">аҫтөркөмө</a> булһа, йәғни <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>-гә ингән теләһә ниндәй ике элементтың суммаһы <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>-гә инә, шулай уҡ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in I\Rightarrow -a\in I.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>I</mi> <mo stretchy="false">⇒</mo> <mo>−</mo> <mi>a</mi> <mo>∈</mo> <mi>I</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in I\Rightarrow -a\in I.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b157cf29b7a0de08aa270141858001ece056295d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.553ex; height:2.343ex;" alt="{\displaystyle a\in I\Rightarrow -a\in I.}"></span></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> ҡулсаның теләһә ниндәй элементына һулдан ҡабатлауға ҡарата йомоҡ, йәғни теләһә ниндәй <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in I,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>I</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in I,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aacb5f25986a2c25f6735c8d24448dbfec5af9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.889ex; height:2.509ex;" alt="{\displaystyle a\in I,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca49c66b5e9b5f32249a737e4429c3df136c33f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.653ex; height:2.176ex;" alt="{\displaystyle r\in R}"></span> өсөн <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ra\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>a</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ra\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dfe76a6a01273724eea2cfd88355369831d4347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.291ex; height:2.176ex;" alt="{\displaystyle ra\in I}"></span>.</li></ul> <p>Беренсе үҙсәнлектән <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> үҙе эсендә ҡабатлауға ҡарата йомоҡ икәне килеп сыға, шуға ла <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> аҫҡулса була. </p><p>Оҡшаш рәүештә ҡулсаның элементына уңдан ҡабатлауға ҡарата йомоҡ уң идеал билдәләмәһе бирелә. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының ике яҡлы идеалы (йәки тик идеал) — бер үк ваҡытта уң да, һул да идеал булған, буш булмаған теләһә ниндәй аҫкүмәклеге. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының идеалына шулай уҡ ниндәйҙер <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to R'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <msup> <mi>R</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:R\to R'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600595876ed2f302630e40f0cc9ffd51fcdc5fa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.043ex; height:2.843ex;" alt="{\displaystyle f:R\to R'}"></span> <a href="/w/index.php?title=%D0%93%D0%BE%D0%BC%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC&action=edit&redlink=1" class="new" title="Гомоморфизм (был бит юҡ)">гомоморфизмының</a> үҙәге тип билдәләмә бирергә мөмкин<sup id="cite_ref-Фейс—1977——153_15-0" class="reference"><a href="#cite_note-Фейс—1977——153-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>. </p><p>Әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының элементы булһа, ул саҡта <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8072625515d9f43b2fd354deb2126199df9bb4cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.094ex; height:2.176ex;" alt="{\displaystyle Rx}"></span> күренешендәге элементтар күмәклеге, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> барлыҡҡа килтергән (ярашлы рәүештә, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xR}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xR}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed2cde95f1c5721ae409a5f7de078d48920300a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.094ex; height:2.176ex;" alt="{\displaystyle xR}"></span>) һул (ярашлы рәүештә, уң) <a href="/w/index.php?title=%D0%A2%D3%A9%D0%BF_%D0%B8%D0%B4%D0%B5%D0%B0%D0%BB&action=edit&redlink=1" class="new" title="Төп идеал (был бит юҡ)">төп идеалы</a> тип атала. Әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһы коммутатив булһа, был билдәләмәләр тап киләләр һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> барлыҡҡа килтергән төп идеал <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1f6d437f1742bfd5ffbdccd2477c07e9909e04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.139ex; height:2.843ex;" alt="{\displaystyle (x)}"></span> тип тамғалана. Мәҫәлән, бөтә йоп һандар күмәклеге бөтөн һандар ҡулсаһында идеал барлыҡҡа килтерә, был идеалды 2 элементы барлыҡҡа килтерә. Бөтөн һандар ҡулсаһында бөтә идеалдар төп идеал була икәнен иҫбат итергә була. <sup id="cite_ref-Куликов—1979——430—431_16-0" class="reference"><a href="#cite_note-Куликов—1979——430—431-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>. </p><p>Ҡулсаның бөтә ҡулса менән тап килмәгән идеалы, әгәр был идеал буйынса <a href="/w/index.php?title=%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Факторҡулса (был бит юҡ)">факторҡулсала</a> нулдең бүлеүселәре булмаһа, <a href="/w/index.php?title=%D0%AF%D0%B1%D0%B0%D0%B9_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Ябай ҡулса (был бит юҡ)">ябай</a> тип атала. Ҡулсаның бөтә ҡулса менән тап килмәгән һәм ҡулсаға тигеҙ булмаған бер ниндәй ҙә ҙурыраҡ идеалға инмәгән идеалы, <a href="/w/index.php?title=%D0%9C%D0%B0%D0%BA%D1%81%D0%B8%D0%BC%D0%B0%D0%BB%D1%8C_%D0%B8%D0%B4%D0%B5%D0%B0%D0%BB&action=edit&redlink=1" class="new" title="Максималь идеал (был бит юҡ)">максималь</a> идеал тип атала<sup id="cite_ref-Винберг—2011——406_17-0" class="reference"><a href="#cite_note-Винберг—2011——406-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Гомоморфизм"><span id=".D0.93.D0.BE.D0.BC.D0.BE.D0.BC.D0.BE.D1.80.D1.84.D0.B8.D0.B7.D0.BC"></span>Гомоморфизм</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=8" title="Гомоморфизм бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=8" title="Гомоморфизм бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ҡулсалар гомоморфизмы (ҡулсалы гомоморфизм) — ҡушыу һәм ҡабатлау операцияларын һаҡлаусы сағылыш ул. Ә атап әйткәндә, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһынан <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> ҡулсаһына <a href="/w/index.php?title=%D0%93%D0%BE%D0%BC%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC&action=edit&redlink=1" class="new" title="Гомоморфизм (был бит юҡ)">гомоморфизм</a> — ул шундай <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:R\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9aa7daf2186fbf2fb3b5090e495ad432d4cf42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle f:R\to S}"></span> <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика)">функцияһы</a>, бында </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a+b)=f(a)+f(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a+b)=f(a)+f(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/822600027fb446b01dd1902d7d3864d9f0f905ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.498ex; height:2.843ex;" alt="{\displaystyle f(a+b)=f(a)+f(b)}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a\cdot b)=f(a)\cdot f(b),~\forall a,b\in ~R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mtext> </mtext> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a\cdot b)=f(a)\cdot f(b),~\forall a,b\in ~R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/433139043f9b2f8c908f7defc847c55b56d2df90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.529ex; height:2.843ex;" alt="{\displaystyle f(a\cdot b)=f(a)\cdot f(b),~\forall a,b\in ~R}"></span>.</li></ol> <p>Берәмеге булған ҡулса осрағында, ҡайһы берҙә <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c23ec03a1dad7631fc47878cb66b800a538dff1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(1)=1}"></span> шарты үтәлеүе талап ителә<sup id="cite_ref-Фейс—1979——10_18-0" class="reference"><a href="#cite_note-Фейс—1979——10-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Винберг—2011——388_19-0" class="reference"><a href="#cite_note-Винберг—2011——388-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>. </p><p>Ҡулсалар гомоморфизмы, әгәр ҡулсаларҙың <a href="/w/index.php?title=%D0%9A%D0%B8%D1%80%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Кире функция (был бит юҡ)">кире</a> гомоморфизмы булһа, изоморфизм тип атала. Ҡулсаларҙың теләһә ниндәй биектив гомоморфизмы изоморфизм була. <a href="/w/index.php?title=%D0%90%D0%B2%D1%82%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC&action=edit&redlink=1" class="new" title="Автоморфизм (был бит юҡ)">Автоморфизм</a> — ҡулсанан үҙенә изоморфизм булған гомоморфизм ул. Миҫал: ҡулсаның үҙенә тождестволы сағылышы автоморфизм була<sup id="cite_ref-Куликов—1979——107—108_20-0" class="reference"><a href="#cite_note-Куликов—1979——107—108-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>. </p><p>Әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:R\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9aa7daf2186fbf2fb3b5090e495ad432d4cf42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle f:R\to S}"></span> — ҡулсалар гомоморфизмы булһа, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-ҙың нолгә күскән элементтары күмәклеге <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>-тың <a href="/w/index.php?title=%D2%AE%D2%99%D3%99%D0%BA_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&action=edit&redlink=1" class="new" title="Үҙәк (алгебра) (был бит юҡ)">үҙәге</a> тип атала (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ker} f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ker} f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2164aff497a3b4d6e25f241b63c86814247941c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.45ex; height:2.509ex;" alt="{\displaystyle \mathrm {ker} f}"></span> тип тамғалана). Теләһә ниндәй гомоморфизмдың үҙәге ике яҡлы идеал була<sup id="cite_ref-Куликов—1979——432_21-0" class="reference"><a href="#cite_note-Куликов—1979——432-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>. Икенсе яҡтан, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>-тың образы һәр ваҡытта ла идеал булмай, ләкин <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>-тың аҫҡулсаһы була <sup id="cite_ref-Фейс—1977——153_15-1" class="reference"><a href="#cite_note-Фейс—1977——153-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {im} f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {im} f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5554206898dfd67a4a133f44e297c27a9014dfe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.861ex; height:2.509ex;" alt="{\displaystyle \mathrm {im} f}"></span> тип тамғалана). </p> <div class="mw-heading mw-heading3"><h3 id="Факторҡулса"><span id=".D0.A4.D0.B0.D0.BA.D1.82.D0.BE.D1.80.D2.A1.D1.83.D0.BB.D1.81.D0.B0"></span>Факторҡулса</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=9" title="Факторҡулса бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=9" title="Факторҡулса бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="dablink noprint">Төп мәҡәлә: <b><a href="/w/index.php?title=%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%BA%D0%BE%D0%BB%D1%8C%D1%86%D0%BE&action=edit&redlink=1" class="new" title="Факторкольцо (был бит юҡ)">Факторкольцо</a></b> </div> <p>Идеал буйынса факторҡулсаның билдәләмәһе <a href="/w/index.php?title=%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D1%82%D3%A9%D1%80%D0%BA%D3%A9%D0%BC&action=edit&redlink=1" class="new" title="Фактортөркөм (был бит юҡ)">фактортөркөм</a> билдәләмәһенә оҡшаш. Теүәлерәк әйткәндә, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының ике яҡлы <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> идеалы буйынса факторҡулсаһы — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> аддитив төркөмөнөң <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> аддитив аҫтөркөмө буйынса, түбәндәге операциялар менән <a href="/w/index.php?title=%D0%99%D3%99%D0%BD%D3%99%D1%88%D3%99%D0%BB%D0%B5%D0%BA_%D0%BA%D0%BB%D0%B0%D1%81%D1%8B&action=edit&redlink=1" class="new" title="Йәнәшәлек класы (был бит юҡ)">йәнәшәлек кластары</a> күмәклеге ул: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+I)+(b+I)=(a+b)+I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+I)+(b+I)=(a+b)+I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7debba6c9ceb70f061211f7be2fb63cba01144e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.698ex; height:2.843ex;" alt="{\displaystyle (a+I)+(b+I)=(a+b)+I}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+I)(b+I)=(ab)+I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+I)(b+I)=(ab)+I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15857e9197fa5b2096c1bb7b959e1e1a151bd15f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.018ex; height:2.843ex;" alt="{\displaystyle (a+I)(b+I)=(ab)+I}"></span>.</li></ul> <p>Төркөмдәр осрағына оҡшаш рәүештә, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x+I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>x</mi> <mo>+</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x+I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996841b61225255019ed51223bca7f25b28a181b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.286ex; height:2.343ex;" alt="{\displaystyle x\mapsto x+I}"></span> тип бирелгән <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:R\to R/I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:R\to R/I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8084fa29746da849ea481b2cce72ba970c9873d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.672ex; height:2.843ex;" alt="{\displaystyle p:R\to R/I}"></span> ҡануни гомоморфизмы бар. Был осраҡта <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> идеалы үҙәге була. </p><p>Төркөмдәр гомоморфизмы тураһында теоремаға оҡшаш рәүештә ҡулсалар гомоморфизмы тураһында теорема бар: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to R'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <msup> <mi>R</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:R\to R'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600595876ed2f302630e40f0cc9ffd51fcdc5fa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.043ex; height:2.843ex;" alt="{\displaystyle f:R\to R'}"></span> булһын, ул саҡта <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Im} f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Im} f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b001b0a33db2c040d74d1114fa459e3416922416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.054ex; height:2.509ex;" alt="{\displaystyle \mathrm {Im} f}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Im} f\simeq A/\mathrm {Ker} f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mi>f</mi> <mo>≃</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Im} f\simeq A/\mathrm {Ker} f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19995841a20b25daac9d39987abeec74529ba79f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.089ex; height:2.843ex;" alt="{\displaystyle \mathrm {Im} f\simeq A/\mathrm {Ker} f}"></span> гомоморфизмы үҙәге буйынса факторҡулсаға изоморфлы <sup id="cite_ref-Винберг—2011——387—390_22-0" class="reference"><a href="#cite_note-Винберг—2011——387—390-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Ҡулсаларҙың_ҡайһы_бер_махсус_кластары"><span id=".D2.A0.D1.83.D0.BB.D1.81.D0.B0.D0.BB.D0.B0.D1.80.D2.99.D1.8B.D2.A3_.D2.A1.D0.B0.D0.B9.D2.BB.D1.8B_.D0.B1.D0.B5.D1.80_.D0.BC.D0.B0.D1.85.D1.81.D1.83.D1.81_.D0.BA.D0.BB.D0.B0.D1.81.D1.82.D0.B0.D1.80.D1.8B"></span>Ҡулсаларҙың ҡайһы бер махсус кластары</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=10" title="Ҡулсаларҙың ҡайһы бер махсус кластары бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=10" title="Ҡулсаларҙың ҡайһы бер махсус кластары бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Һәр нулдән айырмалы элементы әйләндермәле булған, берәмеге <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bee82995e4ac724a7b5fdfb4b0d76560321e1d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.676ex;" alt="{\displaystyle 1\neq 0}"></span> булған ҡулса <a href="/w/index.php?title=%D0%95%D1%81%D0%B5%D0%BC_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&action=edit&redlink=1" class="new" title="Есем (алгебра) (был бит юҡ)">есем</a> тип атала<sup id="cite_ref-Винберг—2011——523_23-0" class="reference"><a href="#cite_note-Винберг—2011——523-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>.</li> <li>Коммутатив есем <a href="/wiki/%D0%AF%D0%BB%D0%B0%D0%BD_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Ялан (алгебра)">ялан</a> тип атала<sup id="cite_ref-Фейс—1977——152_24-0" class="reference"><a href="#cite_note-Фейс—1977——152-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>; икенсе төрлө әйткәндә, ялан — тривиаль булмаған <a href="/w/index.php?title=%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&action=edit&redlink=1" class="new" title="Идеал (алгебра) (был бит юҡ)">идеалдары</a> булмаған, берәмеге булған коммутатив ҡулса<sup id="cite_ref-Атья,_Макдональд—1972——11_8-1" class="reference"><a href="#cite_note-Атья,_Макдональд—1972——11-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Куликов—1979——430_25-0" class="reference"><a href="#cite_note-Куликов—1979——430-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>.</li> <li>Нулдең бүлеүселәре булмаған коммутатив ҡулса <a href="/w/index.php?title=%D0%91%D3%A9%D1%82%D3%A9%D0%BD%D0%BB%D3%A9%D0%BA_%D3%A9%D0%BB%D0%BA%D3%99%D2%BB%D0%B5&action=edit&redlink=1" class="new" title="Бөтөнлөк өлкәһе (был бит юҡ)">бөтөнлөк өлкәһе</a> (йәки бөтөн ҡулса) тип атала<sup id="cite_ref-Винберг—2011——118_26-0" class="reference"><a href="#cite_note-Винберг—2011——118-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>. Теләһә ниндәй ялан бөтөнлөк өлкәһе була, ләкин киреһе дөрөҫ түгел<sup id="cite_ref-Атья,_Макдональд—1972——_27-0" class="reference"><a href="#cite_note-Атья,_Макдональд—1972——-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>.</li> <li>Ялан булмаған <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> бөтөн ҡулса <a href="/w/index.php?title=%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Евклид ҡулсаһы (был бит юҡ)">Евклид ҡулсаһы</a> тип атала, әгәр ҡулсала шундай <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\colon R\to Z_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\colon R\to Z_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d59d35b0252cd0fa1e115ffaa487be5bb34d79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.574ex; height:2.509ex;" alt="{\displaystyle N\colon R\to Z_{+}}"></span> нормаһы бирелһә, бында: <ol><li>теләһә ниндәй нулдән айырмалы <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d992800adce8f580e87a3c79b62f0e12d5349e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.509ex;" alt="{\displaystyle a,b\in R}"></span> өсөн, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(a)\leq N(ab)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(a)\leq N(ab)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efa25143756b81ed94a982ec1b60dc7219992bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.301ex; height:2.843ex;" alt="{\displaystyle N(a)\leq N(ab)}"></span> дөрөҫ;</li> <li>теләһә ниндәй нулдән айырмалы <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d992800adce8f580e87a3c79b62f0e12d5349e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.509ex;" alt="{\displaystyle a,b\in R}"></span> өсөн шундай <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q,r\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q,r\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3542c28e1c0a6135378617185ba7698a8de865c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.757ex; height:2.509ex;" alt="{\displaystyle q,r\in R}"></span> бар, бында <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=qb+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>q</mi> <mi>b</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=qb+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a579e5586f4c6c9bbc5dc12f533988db18a833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.284ex; height:2.509ex;" alt="{\displaystyle a=qb+r}"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"></span> йәки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(r)<N(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(r)<N(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd590cc1a84e439235097cd7d0ba38ee74bb5211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.89ex; height:2.843ex;" alt="{\displaystyle N(r)<N(b)}"></span><sup id="cite_ref-Винберг—2011——118_26-1" class="reference"><a href="#cite_note-Винберг—2011——118-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>.</li></ol></li> <li>Бөтә <a href="/w/index.php?title=%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&action=edit&redlink=1" class="new" title="Идеал (алгебра) (был бит юҡ)">идеалдары</a> ла төп идеал булған бөтөн ҡулса <a href="/w/index.php?title=%D0%A2%D3%A9%D0%BF_%D0%B8%D0%B4%D0%B5%D0%B0%D0%BB%D0%B4%D0%B0%D1%80_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Төп идеалдар ҡулсаһы (был бит юҡ)">төп идеалдар ҡулсаһы</a> тип атала; һәр Евклид ҡулсаһы һәм һәр ялан төп идеалдар ҡулсаһы булалар<sup id="cite_ref-Винберг—2011——21_12-1" class="reference"><a href="#cite_note-Винберг—2011——21-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>.</li> <li><span id=".D2.BA.D0.B0.D0.BD.D0.BB.D1.8B_.D2.A1.D1.83.D0.BB.D1.81.D0.B0"><span id="Һанлы_ҡулса"></span></span>Элементтары булып <a href="/wiki/%D2%BA%D0%B0%D0%BD" title="Һан">һандар</a> торған, ә операциялары — һандарҙы <a href="/wiki/%D2%A0%D1%83%D1%88%D1%8B%D1%83" title="Ҡушыу">ҡушыу</a> һәм <a href="/wiki/%D2%A0%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D1%83" title="Ҡабатлау">ҡабатлау</a> булған ҡулса <i><b>һанлы ҡулса</b></i> тип атала, мәҫәлән, йоп һандар күмәклеге һанлы ҡулса була, ләкин бер ниндәй ҙә тиҫкәре һандар системаһы ҡулса булмай, сөнки уларҙың ҡабатландығы ыңғай<sup id="cite_ref-Курош—1968——266_28-0" class="reference"><a href="#cite_note-Курош—1968——266-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Миҫалдар"><span id=".D0.9C.D0.B8.D2.AB.D0.B0.D0.BB.D0.B4.D0.B0.D1.80"></span>Миҫалдар</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=11" title="Миҫалдар бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=11" title="Миҫалдар бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}"></span> — бер нулдән генә торған <a href="/w/index.php?title=%D0%A2%D1%80%D0%B8%D0%B2%D0%B8%D0%B0%D0%BB%D1%8C_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Тривиаль ҡулса (был бит юҡ)">тривиаль ҡулса</a> булһын. Был нол мультипликатив берәмек булған берҙән бер ҡулса<sup id="cite_ref-Винберг—2011——18—19_5-2" class="reference"><a href="#cite_note-Винберг—2011——18—19-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>. Был тривиаль миҫалды <a href="/w/index.php?title=%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F%D0%BB%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Категориялар теорияһы (был бит юҡ)">категориялар теорияһы</a> күҙлегенән ҡарағанда ҡулса тип иҫәпләү файҙалы, сөнки был осраҡта ҡулсалар категорияларында <a href="/w/index.php?title=%D0%A2%D0%B5%D1%80%D0%BC%D0%B8%D0%BD%D0%B0%D0%BB%D1%8C_%D0%BE%D0%B1%D1%8A%D0%B5%D0%BA%D1%82&action=edit&redlink=1" class="new" title="Терминаль объект (был бит юҡ)">терминаль объект</a> барлыҡҡа килә.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> — <a href="/w/index.php?title=%D0%91%D3%A9%D1%82%D3%A9%D0%BD_%D2%BB%D0%B0%D0%BD%D0%B4%D0%B0%D1%80&action=edit&redlink=1" class="new" title="Бөтөн һандар (был бит юҡ)">бөтөн һандар</a> (ғәҙәттәге ҡушыу һәм ҡабатлау менән) ҡулсаһы. Был ҡулсаларҙың бик мөһим миҫалы, сөнки теләһә ниндәй ҡулсаны <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> өҫтөндә <a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_%D3%A9%D2%AB%D1%82%D3%A9%D0%BD%D0%B4%D3%99_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0&action=edit&redlink=1" class="new" title="Ҡулса өҫтөндә алгебра (был бит юҡ)">алгебра</a> итеп ҡарап була. Шулай уҡ ул <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><b>Ring</b></span> берәмек менән ҡулсалар категорияһында <a href="/w/index.php?title=%D0%91%D0%B0%D1%88%D0%BB%D0%B0%D0%BD%D2%93%D1%8B%D1%81_%D0%BE%D0%B1%D1%8A%D0%B5%D0%BA%D1%82&action=edit&redlink=1" class="new" title="Башланғыс объект (был бит юҡ)">башланғыс объект</a> булып тора.<sup id="cite_ref-Фейс—1977——_29-0" class="reference"><a href="#cite_note-Фейс—1977——-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Фейс—1979——_30-0" class="reference"><a href="#cite_note-Фейс—1979——-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b729c334a9863c47f0b7e3ad61342c2f0881bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{n}}"></span> — <span class="math-template" style="font-style:italic;">n</span> натураль һанының модуле буйынса <a href="/wiki/%D0%9C%D0%BE%D0%B4%D1%83%D0%BB%D0%B5_%D0%B1%D1%83%D0%B9%D1%8B%D0%BD%D1%81%D0%B0_%D1%81%D0%B0%D2%93%D1%8B%D1%88%D1%82%D1%8B%D1%80%D1%8B%D1%83" title="Модуле буйынса сағыштырыу">вычеттар</a> <a href="/w/index.php?title=%D0%A1%D0%B8%D0%BA%D0%BB%D0%B5_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Сикле ҡулса (был бит юҡ)">сикле ҡулсаһы</a>. Был һандар теорияһынан ҡулсаларҙың классик миҫалы. Вычеттар ҡулсаһы ялан була шул саҡта һәм бары тик шул саҡта ғына, әгәр <span class="math-template" style="font-style:italic;">n</span> <a href="/wiki/%D0%AF%D0%B1%D0%B0%D0%B9_%D2%BB%D0%B0%D0%BD" title="Ябай һан">ябай</a> булһа.<sup id="cite_ref-Винберг—2011——28—34_31-0" class="reference"><a href="#cite_note-Винберг—2011——28—34-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Ярашлы яландар <a href="/w/index.php?title=%D0%A1%D0%B8%D0%BA%D0%BB%D0%B5_%D1%8F%D0%BB%D0%B0%D0%BD&action=edit&redlink=1" class="new" title="Сикле ялан (был бит юҡ)">сикле яландар</a> теорияһын төҙөү өсөн башланғыс нөктә булып торалар. Вычеттар ҡулсалары шулай уҡ <a href="/w/index.php?title=%D0%A1%D0%B8%D0%BA%D0%BB%D0%B5_%D0%B1%D0%B0%D1%80%D0%BB%D1%8B%D2%A1%D2%A1%D0%B0_%D0%BA%D0%B8%D0%BB%D1%82%D0%B5%D1%80%D0%B5%D0%BB%D0%B3%D3%99%D0%BD_%D0%90%D0%B1%D0%B5%D0%BB%D1%8C_%D1%82%D3%A9%D1%80%D0%BA%D3%A9%D0%BC%D3%A9&action=edit&redlink=1" class="new" title="Сикле барлыҡҡа килтерелгән Абель төркөмө (был бит юҡ)">сикле барлыҡҡа килтерелгән Абель төркөмдәре</a> структураларын өйрәнгәндә бик мөһим, уларҙы шулай уҡ <a href="/w/index.php?title=P-%D0%B0%D0%B4%D0%B8%D0%BA%D0%BB%D1%8B_%D2%BB%D0%B0%D0%BD&action=edit&redlink=1" class="new" title="P-адиклы һан (был бит юҡ)"><span class="math-template" style="font-style:italic;">p</span>-адиклы һандарҙы</a> төҙөү өсөн ҡулланырға була.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> — ялан булып торған <a href="/wiki/%D0%A0%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C_%D2%BB%D0%B0%D0%BD" title="Рациональ һан">рациональ һандар</a> ҡулсаһы. Был <a href="/w/index.php?title=%D0%AF%D0%BB%D0%B0%D0%BD%D0%B4%D1%8B%D2%A3_%D1%85%D0%B0%D1%80%D0%B0%D0%BA%D1%82%D0%B5%D1%80%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Яландың характеристикаһы (был бит юҡ)"> характеристикаһы</a> 0 булған иң ябай ялан. Ул һандар теорияһында төп өйрәнеү объекты булып тора. Уны төрлө эквивалентлы булмаған нормалар буйынса тултырыу <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> <a href="/wiki/%D0%AB%D1%81%D1%8B%D0%BD_%D2%BB%D0%B0%D0%BD" title="Ысын һан">ысын һандар</a> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d194e3e8fce9335ed524db967666b4f02fb523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.514ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p},}"></span> бында <span class="math-template" style="font-style:italic;">p</span> — теләһә ниндәй ябай һан <span class="math-template" style="font-style:italic;">p</span>-адиклы һандар яландарын бирә<sup id="cite_ref-Ван_дер_Варден—1975——509—512_32-0" class="reference"><a href="#cite_note-Ван_дер_Варден—1975——509—512-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup>.</li> <li>Теләһә ниндәй <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> коммутатив ҡулсаһы өсөн коэффициенттары <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ысын һандар булған <span class="math-template" style="font-style:italic;">n</span> үҙгәреүсәнле <a href="/w/index.php?title=%D0%9A%D2%AF%D0%BF%D0%B1%D1%8B%D1%83%D1%8B%D0%BD%D0%B4%D0%B0%D1%80_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Күпбыуындар ҡулсаһы (был бит юҡ)">күпбыуындар ҡулсаһы</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[x_{1},x_{2},\dots ,x_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x_{1},x_{2},\dots ,x_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de1d5767a8f5ee8e1b21bad9b0a1b973342f0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.586ex; height:2.843ex;" alt="{\displaystyle R[x_{1},x_{2},\dots ,x_{n}]}"></span> төҙөргә мөмкин.<sup id="cite_ref-Куликов—1979——110—111_11-2" class="reference"><a href="#cite_note-Куликов—1979——110—111-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Айырым алғанда, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[x][y]=R[x,y].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x][y]=R[x,y].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/634dbdfb9abac4cc6d44e11c3b5aafc4a37c73d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.159ex; height:2.843ex;" alt="{\displaystyle R[x][y]=R[x,y].}"></span> Коэффициенттары бөтөн һандар булған күпбыуындар ҡулсалары, бөтә күпбыуындар ҡулсаларының <a href="/w/index.php?title=%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80%D0%BB%D1%8B_%D2%A1%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D0%BD%D0%B4%D1%8B%D2%A1&action=edit&redlink=1" class="new" title="Тензорлы ҡабатландыҡ (был бит юҡ)">тензорлы ҡабатландығы</a> аша күрһәтеләләр тигән мәғәнәлә, универсаль күпбыуындар ҡулсаһы булалар: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[x_{1},\dots ,x_{n}]=R\otimes \left(\mathbb {Z} [x_{1},\dots ,x_{n}]\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>R</mi> <mo>⊗</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x_{1},\dots ,x_{n}]=R\otimes \left(\mathbb {Z} [x_{1},\dots ,x_{n}]\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7578a64a6b1b1ca956a10f00edd7d510f42ce683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.668ex; height:2.843ex;" alt="{\displaystyle R[x_{1},\dots ,x_{n}]=R\otimes \left(\mathbb {Z} [x_{1},\dots ,x_{n}]\right).}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> күмәклегенең аҫкүмәклектәре ҡулсаһы — элементтары булып <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-тың аҫкүмәклектәре торған ҡулса. Ҡушыу операцияһы <a href="/w/index.php?title=%D0%A1%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BA_%D0%B0%D0%B9%D1%8B%D1%80%D0%BC%D0%B0&action=edit&redlink=1" class="new" title="Симметрик айырма (был бит юҡ)">симметрик айырма</a>, ә ҡабатлау — <a href="/w/index.php?title=%D0%9A%D2%AF%D0%BC%D3%99%D0%BA%D0%BB%D0%B5%D0%BA%D1%82%D3%99%D1%80%D2%99%D0%B5%D2%A3_%D0%BA%D0%B8%D2%AB%D0%B5%D0%BB%D0%B5%D1%88%D0%B5&action=edit&redlink=1" class="new" title="Күмәклектәрҙең киҫелеше (был бит юҡ)">күмәклектәрҙең киҫелеше</a>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B=A\Delta B=(A\setminus B)\cup (B\setminus A),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>=</mo> <mi>A</mi> <mi mathvariant="normal">Δ</mi> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B=A\Delta B=(A\setminus B)\cup (B\setminus A),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e542068833a2b0ef1ac1da0ed27b1f79b4cf2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.239ex; height:2.843ex;" alt="{\displaystyle A+B=A\Delta B=(A\setminus B)\cup (B\setminus A),}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot B=A\cap B.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> <mo>=</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot B=A\cap B.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571d967973d1101c8478ba7311199e6a0f4b706b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.021ex; height:2.176ex;" alt="{\displaystyle A\cdot B=A\cap B.}"></span></dd> <dd>Ҡулсаның аксиомаларын еңел тикшереп була. Нуль элемент булып буш күмәклек тора, берәмек — бөтә <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"></span> Ҡулсаның бөтә элементтары идемпотента булалар, йәғни <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot A=A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot A=A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3c6319882128add0d13ff7b9fbaddf0dadf507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.654ex; height:2.176ex;" alt="{\displaystyle A\cdot A=A.}"></span> Теләһә ниндәй элемент ҡушыу буйынса үҙенә кире элемент була: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+A=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>A</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+A=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2253f11b744789a5ac0e87b48f72951be7fb14fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.234ex; height:2.343ex;" alt="{\displaystyle A+A=0.}"></span> Аҫкүмәклектәре ҡулсаһы <a href="/wiki/%D0%91%D1%83%D0%BB%D1%8C_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D2%BB%D1%8B" title="Буль алгебраһы">Буль алгебралары</a> теорияһында һәм <a href="/w/index.php?title=%D0%9A%D2%AF%D0%BC%D3%99%D0%BA%D0%BB%D0%B5%D0%BA_%D2%AF%D0%BB%D1%81%D3%99%D0%BC%D0%B5&action=edit&redlink=1" class="new" title="Күмәклек үлсәме (был бит юҡ)">үлсәмдәр теорияһында</a> бик мөһим, айырым алғанда <a href="/wiki/%D0%98%D1%85%D1%82%D0%B8%D0%BC%D0%B0%D0%BB%D0%BB%D1%8B%D2%A1_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Ихтималлыҡ теорияһы">ихтималлыҡ теорияһын</a> төҙөгәндә<sup id="cite_ref-Винберг—2011——18—19_5-3" class="reference"><a href="#cite_note-Винберг—2011——18—19-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Конструкциялар"><span id=".D0.9A.D0.BE.D0.BD.D1.81.D1.82.D1.80.D1.83.D0.BA.D1.86.D0.B8.D1.8F.D0.BB.D0.B0.D1.80"></span>Конструкциялар</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=12" title="Конструкциялар бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=12" title="Конструкциялар бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Тура_ҡабатландыҡ"><span id=".D0.A2.D1.83.D1.80.D0.B0_.D2.A1.D0.B0.D0.B1.D0.B0.D1.82.D0.BB.D0.B0.D0.BD.D0.B4.D1.8B.D2.A1"></span>Тура ҡабатландыҡ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=13" title="Тура ҡабатландыҡ бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=13" title="Тура ҡабатландыҡ бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> ҡулсаларының <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\times S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>×</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\times S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c3ff61b6cca09ae2b3fb47ba9417b51d83b94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.104ex; height:2.176ex;" alt="{\displaystyle R\times S}"></span> <a href="/w/index.php?title=%D0%A2%D1%83%D1%80%D0%B0_%D2%A1%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D0%BD%D0%B4%D1%8B%D2%A1&action=edit&redlink=1" class="new" title="Тура ҡабатландыҡ (был бит юҡ)">ҡабатландығын</a> тәбиғи ҡулса структураһы менән тәьмин итергә мөмкин: теләһә ниндәй <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1},r_{2}\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1},r_{2}\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/249c192b1c75f367a153c6e3dd916cd7f6a101b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.844ex; height:2.509ex;" alt="{\displaystyle r_{1},r_{2}\in R}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1},s_{2}\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{1},s_{2}\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8498313eb8ef2c1631a386a7dff8f4fb992619e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.663ex; height:2.509ex;" alt="{\displaystyle s_{1},s_{2}\in S}"></span> өсөн: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{1},s_{1})+(r_{2},s_{2})=(r_{1}+r_{2},s_{1}+s_{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{1},s_{1})+(r_{2},s_{2})=(r_{1}+r_{2},s_{1}+s_{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f6cf2c635d9816a7fd4be0a1d569d7b818aa0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.787ex; height:2.843ex;" alt="{\displaystyle (r_{1},s_{1})+(r_{2},s_{2})=(r_{1}+r_{2},s_{1}+s_{2}),}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{1},s_{1})\cdot (r_{2},s_{2})=(r_{1}r_{2},s_{1}s_{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{1},s_{1})\cdot (r_{2},s_{2})=(r_{1}r_{2},s_{1}s_{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc0fb8e01d17020689dd136ee599c052e880498f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.944ex; height:2.843ex;" alt="{\displaystyle (r_{1},s_{1})\cdot (r_{2},s_{2})=(r_{1}r_{2},s_{1}s_{2}).}"></span></li></ul> <p>Оҡшаш конструкция теләһә ниндәй ҡулсалар ғәиләһе ҡабатландығы өсөн бар (ҡушыу һәм ҡабатлау компоненттар буйынса биреләләр)<sup id="cite_ref-Ван_дер_Варден—1975——33_33-0" class="reference"><a href="#cite_note-Ван_дер_Варден—1975——33-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> — <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2_%D2%A1%D1%83%D0%BB%D1%81%D0%B0" title="Коммутатив ҡулса">коммутатив ҡулса</a> һәм <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{1},\cdots ,{\mathfrak {a}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{1},\cdots ,{\mathfrak {a}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9454ea2b80c229917fa96121807f90e4476ec6b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.776ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {a}}_{1},\cdots ,{\mathfrak {a}}_{n}}"></span> — унда пар-пар үҙ-ара ябай идеалдар булһын (идеалдар, әгәр уларҙың <a href="/w/index.php?title=%D0%98%D0%B4%D0%B5%D0%B0%D0%BB%D0%B4%D0%B0%D1%80_%D1%81%D1%83%D0%BC%D0%BC%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Идеалдар суммаһы (был бит юҡ)">суммаһы</a> бөтә ҡулсаға тигеҙ булһа, үҙ-ара ябай тип атала). <a href="/w/index.php?title=%D2%A0%D0%B0%D0%BB%D0%B4%D1%8B%D2%A1%D1%82%D0%B0%D1%80_%D1%82%D1%83%D1%80%D0%B0%D2%BB%D1%8B%D0%BD%D0%B4%D0%B0_%D2%A0%D1%8B%D1%82%D0%B0%D0%B9_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Ҡалдыҡтар тураһында Ҡытай теоремаһы (был бит юҡ)">Ҡалдыҡтар тураһында Ҡытай теоремаһы</a>,: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to R/{\mathfrak {a}}_{1}\times \cdots \times R/{\mathfrak {a}}_{n},\quad x\mapsto (x+{\mathfrak {a}}_{1},\ldots ,x+{\mathfrak {a}}_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">→</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>x</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\to R/{\mathfrak {a}}_{1}\times \cdots \times R/{\mathfrak {a}}_{n},\quad x\mapsto (x+{\mathfrak {a}}_{1},\ldots ,x+{\mathfrak {a}}_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23eba3abb068b3af6d9ce3913ef86dc30bcb6d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.458ex; height:2.843ex;" alt="{\displaystyle R\to R/{\mathfrak {a}}_{1}\times \cdots \times R/{\mathfrak {a}}_{n},\quad x\mapsto (x+{\mathfrak {a}}_{1},\ldots ,x+{\mathfrak {a}}_{n})}"></span> сағылышы сюръектив, ә уның үҙәге — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod {\mathfrak {a}}_{i}=\cap {\mathfrak {a}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∏</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>∩</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod {\mathfrak {a}}_{i}=\cap {\mathfrak {a}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0093d95b4ae831041aa8a169d4311a7870263eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.93ex; height:3.843ex;" alt="{\displaystyle \prod {\mathfrak {a}}_{i}=\cap {\mathfrak {a}}_{i}}"></span> (<a href="/w/index.php?title=%D0%98%D0%B4%D0%B5%D0%B0%D0%BB%D0%B4%D0%B0%D1%80_%D2%A1%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D0%BD%D0%B4%D1%8B%D2%93%D1%8B&action=edit&redlink=1" class="new" title="Идеалдар ҡабатландығы (был бит юҡ)">идеалдар ҡабатландығы</a>, <a href="/w/index.php?title=%D0%98%D0%B4%D0%B5%D0%B0%D0%BB%D0%B4%D0%B0%D1%80_%D0%BA%D0%B8%D2%AB%D0%B5%D0%BB%D0%B5%D1%88%D0%B5&action=edit&redlink=1" class="new" title="Идеалдар киҫелеше (был бит юҡ)">идеалдар киҫелеше</a>) тип раҫлай<sup id="cite_ref-Фейс—1979——10_18-1" class="reference"><a href="#cite_note-Фейс—1979——10-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Эндоморфизмдар_ҡулсаһы"><span id=".D0.AD.D0.BD.D0.B4.D0.BE.D0.BC.D0.BE.D1.80.D1.84.D0.B8.D0.B7.D0.BC.D0.B4.D0.B0.D1.80_.D2.A1.D1.83.D0.BB.D1.81.D0.B0.D2.BB.D1.8B"></span>Эндоморфизмдар ҡулсаһы</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=14" title="Эндоморфизмдар ҡулсаһы бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=14" title="Эндоморфизмдар ҡулсаһы бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc1aea157994289d7d45060934d3815375eab3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle (A,+)}"></span> <a href="/wiki/%D0%90%D0%B1%D0%B5%D0%BB%D1%8C_%D1%82%D3%A9%D1%80%D0%BA%D3%A9%D0%BC%D3%A9" title="Абель төркөмө">Абель төркөмөнөң</a> <a href="/w/index.php?title=%D0%AD%D0%BD%D0%B4%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC&action=edit&redlink=1" class="new" title="Эндоморфизм (был бит юҡ)">эндоморфизмдар</a> күмәклеге ҡулса төҙөй, ул <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {End} (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>End</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {End} (A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a15559fa07f2d6c70802f1f3ca215c9a2782c21e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.72ex; height:2.843ex;" alt="{\displaystyle \operatorname {End} (A)}"></span> тип тамғалана. Ике эндомрофризмдың суммаһы компоненттар буйынса билдәләнә: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)(x)=f(x)+g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f+g)(x)=f(x)+g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf80cb50218eac1e40d4a0908bd039db3bd0863c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.795ex; height:2.843ex;" alt="{\displaystyle (f+g)(x)=f(x)+g(x)}"></span>, ә ҡабатландығы — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (fg)(x)=f(g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (fg)(x)=f(g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac54124afb8af70b0b7612faf4bde00796e9e63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.784ex; height:2.843ex;" alt="{\displaystyle (fg)(x)=f(g(x))}"></span> композицияһы кеүек. Әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc1aea157994289d7d45060934d3815375eab3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle (A,+)}"></span> — Абель төркөмө булмаһа, ул саҡта <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d94a24abd865f6f9fd67a7df7e531cae1c769b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f+g}"></span>, дөйөм әйткәндә, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g+f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>+</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g+f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a76216d317c2a233fe1d1b77e663152218d3c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle g+f}"></span>-ға тигеҙ түгел, ә ҡулсала ҡушыу коммутатив булырға тейеш<sup id="cite_ref-Ван_дер_Варден—1975——173_34-0" class="reference"><a href="#cite_note-Ван_дер_Варден—1975——173-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Бүлендектәр_яланы_һәм_бүлендектәр_ҡулсаһы"><span id=".D0.91.D2.AF.D0.BB.D0.B5.D0.BD.D0.B4.D0.B5.D0.BA.D1.82.D3.99.D1.80_.D1.8F.D0.BB.D0.B0.D0.BD.D1.8B_.D2.BB.D3.99.D0.BC_.D0.B1.D2.AF.D0.BB.D0.B5.D0.BD.D0.B4.D0.B5.D0.BA.D1.82.D3.99.D1.80_.D2.A1.D1.83.D0.BB.D1.81.D0.B0.D2.BB.D1.8B"></span>Бүлендектәр яланы һәм бүлендектәр ҡулсаһы</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=15" title="Бүлендектәр яланы һәм бүлендектәр ҡулсаһы бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=15" title="Бүлендектәр яланы һәм бүлендектәр ҡулсаһы бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="dablink noprint">Төп мәҡәләләр: <b><a href="/w/index.php?title=%D0%9F%D0%BE%D0%BB%D0%B5_%D1%87%D0%B0%D1%81%D1%82%D0%BD%D1%8B%D1%85&action=edit&redlink=1" class="new" title="Поле частных (был бит юҡ)">Поле частных</a></b>, <b><a href="/w/index.php?title=%D0%9A%D0%BE%D0%BB%D1%8C%D1%86%D0%BE_%D1%87%D0%B0%D1%81%D1%82%D0%BD%D1%8B%D1%85&action=edit&redlink=1" class="new" title="Кольцо частных (был бит юҡ)">Кольцо частных</a></b> </div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> <a href="/w/index.php?title=%D0%91%D3%A9%D1%82%D3%A9%D0%BD_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Бөтөн ҡулса (был бит юҡ)">бөтөн ҡулсаһы</a> өсөн, уны индергән, иң бәләкәй <a href="/wiki/%D0%AF%D0%BB%D0%B0%D0%BD_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Ялан (алгебра)">ялан</a> төҙөргә мөмкинлек биреүсе конструкция бар. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ҡулсаһының <a href="/w/index.php?title=%D0%91%D2%AF%D0%BB%D0%B5%D0%BD%D0%B4%D0%B5%D0%BA%D1%82%D3%99%D1%80_%D1%8F%D0%BB%D0%B0%D0%BD%D1%8B&action=edit&redlink=1" class="new" title="Бүлендектәр яланы (был бит юҡ)">бүлендектәр яланы</a> — түбәндәге <a href="/wiki/%D0%AD%D0%BA%D0%B2%D0%B8%D0%B2%D0%B0%D0%BB%D0%B5%D0%BD%D1%82%D0%BB%D1%8B%D2%A1_%D0%B1%D3%99%D0%B9%D0%BB%D3%99%D0%BD%D0%B5%D1%88%D0%B5" title="Эквивалентлыҡ бәйләнеше">эквивалентлыҡ бәйләнеше</a> буйынса формаль кәсерҙәрҙең <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q,\;p,q\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p/q,\;p,q\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e479593b190aa1ce86cfa20a5ecd6d87eb6978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.048ex; height:2.843ex;" alt="{\displaystyle p/q,\;p,q\in R}"></span> <a href="/w/index.php?title=%D0%AD%D0%BA%D0%B2%D0%B8%D0%B2%D0%B0%D0%BB%D0%B5%D0%BD%D1%82%D0%BD%D0%BB%D1%8B%D2%A1_%D0%BA%D0%BB%D0%B0%D1%81%D1%8B&action=edit&redlink=1" class="new" title="Эквивалентнлыҡ класы (был бит юҡ)">эквивалентлыҡ кластары</a> күмәклеге: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {p_{1} \over q_{1}}\sim {p_{2} \over q_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {p_{1} \over q_{1}}\sim {p_{2} \over q_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87ab4bb9328c45d626722bd05c3a45d8872410b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.218ex; height:5.343ex;" alt="{\displaystyle {p_{1} \over q_{1}}\sim {p_{2} \over q_{2}}}"></span> шул саҡта һәм тик шул саҡта ғына, әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {p_{1}q_{2}}={p_{2}q_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {p_{1}q_{2}}={p_{2}q_{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5656a57c8c214b868e4bda4be1febd0c19609401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.818ex; height:2.009ex;" alt="{\displaystyle {p_{1}q_{2}}={p_{2}q_{1}}}"></span> булһа,</dd></dl> <p>ғәҙәттәге: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a \over b}+{c \over d}={ad+bc \over bd},\quad {a \over b}\cdot {c \over d}={ac \over bd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mi>d</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a \over b}+{c \over d}={ad+bc \over bd},\quad {a \over b}\cdot {c \over d}={ac \over bd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9cc5a7bada74e1c6f6a6dae0227d569ddd3c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.508ex; height:5.509ex;" alt="{\displaystyle {a \over b}+{c \over d}={ad+bc \over bd},\quad {a \over b}\cdot {c \over d}={ac \over bd}}"></span> операциялары менән. </p><p>Бирелгән бәйләнеш ысынлап та эквивалентлыҡ бәйләнеше булыуы асыҡтан-асыҡ күренеп тормай: иҫбатлау өсөн ҡулсаның бөтөнлөгө менән файҙаланырға тура килә. Был конструкцияның ирекле коммутатив ҡулсаларға дөйөмләштерелеүе бар. Атап әйткәндә, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> коммутатив ҡулсаһында <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> мультипликатив йомоҡ система (йәғни берәмеге булған һәм нуле булмаған аҫкүмәклек; аҫкүмәклектең теләһә ниндәй ике элементының ҡабатландығы яңынан аҫкүмәклеккә инә). Ул саҡта <a href="/w/index.php?title=%D0%91%D2%AF%D0%BB%D0%B5%D0%BD%D0%B4%D0%B5%D0%BA%D1%82%D3%99%D1%80_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Бүлендектәр ҡулсаһы (был бит юҡ)">бүлендектәр ҡулсаһы</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36f81aa8006deb333465f99113727ab38fa80e04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}"></span> — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r/s,\;r\in R,s\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>s</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>r</mi> <mo>∈</mo> <mi>R</mi> <mo>,</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r/s,\;r\in R,s\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05522a9f133fcdb54236897d334e02b7f4d2ad1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.098ex; height:2.843ex;" alt="{\displaystyle r/s,\;r\in R,s\in S}"></span> формаль кәсерҙәренең түбәндәге эквивалентлыҡ бәйләнеше буйынса эквивалентлыҡ кластары күмәклеге: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {r_{1} \over s_{1}}\sim {r_{2} \over s_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {r_{1} \over s_{1}}\sim {r_{2} \over s_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05561a9fa7b29a5d6dbf324b32c3dd212f37ce90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.06ex; height:5.009ex;" alt="{\displaystyle {r_{1} \over s_{1}}\sim {r_{2} \over s_{2}}}"></span> шул саҡта һәм тик шул саҡта ғына, әгәр <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s'{r_{1}s_{2}-r_{2}s_{1}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s'{r_{1}s_{2}-r_{2}s_{1}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d45bf897206f5193d7bb20c44034cae491ce8f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.372ex; height:2.843ex;" alt="{\displaystyle s'{r_{1}s_{2}-r_{2}s_{1}}=0}"></span> булған <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s'\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s'\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612f681a39e09dedbb155f8b4e2152dd1d09816d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.115ex; height:2.509ex;" alt="{\displaystyle s'\in S}"></span> булһа.</dd></dl> <p>Шулай уҡ был конструкцияны <a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0%D0%BD%D1%8B_%D0%BB%D0%BE%D0%BA%D0%B0%D0%BB%D0%BB%D3%99%D1%88%D1%82%D0%B5%D1%80%D0%B5%D2%AF&action=edit&redlink=1" class="new" title="Ҡулсаны локалләштереү (был бит юҡ)">ҡулсаны локалләштереү</a> тип атайҙар (сөнки <a href="/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Алгебраик геометрия">алгебраик геометрияла</a> ул <a href="/w/index.php?title=%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D1%82%D3%A9%D1%80%D0%BB%D3%A9%D0%BB%D3%A9%D0%BA&action=edit&redlink=1" class="new" title="Алгебраик төрлөлөк (был бит юҡ)">төрлөлөктөң</a> айырым нөктәһендә локаль үҙсәнлектәрен өйрәнергә мөмкинлек бирә). Миҫал: <a href="/w/index.php?title=%D0%A3%D0%BD%D0%B0%D1%80%D0%BB%D1%8B_%D0%BA%D3%99%D1%81%D0%B5%D1%80&action=edit&redlink=1" class="new" title="Унарлы кәсер (был бит юҡ)">унарлы кәсерҙәр</a> ҡулсаһы — бөтөн һандар ҡулсаһын <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{10^{n}|n\geqslant 0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo>⩾</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{10^{n}|n\geqslant 0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/748c533a659098bfdcd65fcbddaae33b83df2dc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.768ex; height:2.843ex;" alt="{\displaystyle S=\{10^{n}|n\geqslant 0\}}"></span> мультипликатив системаһы буйынса локалләштереү ул. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to S^{-1}R,\,r\mapsto r/1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>R</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>r</mi> <mo stretchy="false">↦</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\to S^{-1}R,\,r\mapsto r/1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9408b4a8c72cf157ce92092c7d2091a5c564700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.454ex; height:3.176ex;" alt="{\displaystyle R\to S^{-1}R,\,r\mapsto r/1}"></span> тәбиғи сағылышы бар. Уның <a href="/w/index.php?title=%D2%AE%D2%99%D3%99%D0%BA_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&action=edit&redlink=1" class="new" title="Үҙәк (алгебра) (был бит юҡ)">үҙәге</a> шундай <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> элементтарынан тора, улар өсөн <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rs=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rs=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/285d1c1df48fd4f79ac656831aff5ed08dd8301f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.4ex; height:2.176ex;" alt="{\displaystyle rs=0}"></span> булырлыҡ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acce52dffd84d073a24f4606a175da60148fd0c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}"></span> бар. Атап әйткәндә, бөтөн ҡулса өсөн был сағылыш <a href="/w/index.php?title=%D0%98%D0%BD%D1%8A%D0%B5%D0%BA%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D2%A1&action=edit&redlink=1" class="new" title="Инъективлыҡ (был бит юҡ)">инъективлы</a><sup id="cite_ref-Ван_дер_Варден—1975——450—452_35-0" class="reference"><a href="#cite_note-Ван_дер_Варден—1975——450—452-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Курош—1968——305—311_36-0" class="reference"><a href="#cite_note-Курош—1968——305—311-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Категориялы_һүрәтләү"><span id=".D0.9A.D0.B0.D1.82.D0.B5.D0.B3.D0.BE.D1.80.D0.B8.D1.8F.D0.BB.D1.8B_.D2.BB.D2.AF.D1.80.D3.99.D1.82.D0.BB.D3.99.D2.AF"></span>Категориялы һүрәтләү</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=16" title="Категориялы һүрәтләү бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=16" title="Категориялы һүрәтләү бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span id=".D0.9A.D0.B0.D1.82.D0.B5.D0.B3.D0.BE.D1.80.D0.BD.D0.BE.D0.B5_.D0.BE.D0.BF.D0.B8.D1.81.D0.B0.D0.BD.D0.B8.D0.B5"><span id="Категорное_описание"></span></span> Ҡулсалар ҡулсалар гомоморфизмдары менән бергә <a href="/w/index.php?title=%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F%D0%BB%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Категориялар теорияһы (был бит юҡ)">категория</a> төҙөйҙәр, ул ғәҙәттә <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Ring} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> <mi mathvariant="bold">i</mi> <mi mathvariant="bold">n</mi> <mi mathvariant="bold">g</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Ring} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e06b22066eb8cbaf751c2ea76f25fee75ee424" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.567ex; height:2.509ex;" alt="{\displaystyle \mathbf {Ring} }"></span> тип тамғалана (ҡайһы берҙә берәмеге булған категорияларҙы шулай тамғалайҙар, ә ғәҙәттәге ҡулсалар категорияларын <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Rng} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> <mi mathvariant="bold">n</mi> <mi mathvariant="bold">g</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Rng} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26eb0f0067e19a801cad3a326d5b5570ec755e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.825ex; height:2.509ex;" alt="{\displaystyle \mathbf {Rng} }"></span> тип тамғалайҙар). Берәмеге булған ҡулсалар категорияһы бик күп файҙалы үҙсәнлектәргә эйә: атап әйткәндә, ул <a href="/w/index.php?title=%D0%A2%D1%83%D0%BB%D1%8B_%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Тулы категория (был бит юҡ)">тулы һәм котулы</a>. Был унда бөтә бәләкәй <a href="/w/index.php?title=%D0%A1%D0%B8%D0%BA%D0%BB%D3%99%D0%BD%D0%BC%D3%99_(%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F%D0%BB%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B)&action=edit&redlink=1" class="new" title="Сикләнмә (категориялар теорияһы) (был бит юҡ)">сикләнмәләрһәм</a> косикләнмәләр (мәҫәлән, <a href="/w/index.php?title=%D2%A0%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D0%BD%D0%B4%D1%8B%D2%A1_(%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F%D0%BB%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B)&action=edit&redlink=1" class="new" title="Ҡабатландыҡ (категориялар теорияһы) (был бит юҡ)">ҡабатландыҡтар</a>, <a href="/w/index.php?title=%D0%9A%D0%BE%D2%A1%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D0%BD%D0%B4%D1%8B%D2%A1&action=edit&redlink=1" class="new" title="Коҡабатландыҡ (был бит юҡ)">коҡабатландыҡтар</a>, <a href="/w/index.php?title=%D2%AE%D2%99%D3%99%D0%BA_(%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F%D0%BB%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B)&action=edit&redlink=1" class="new" title="Үҙәк (категориялар теорияһы) (был бит юҡ)">үҙәктәр</a> һәм <a href="/w/index.php?title=%D0%9A%D0%BE%D2%AF%D2%99%D3%99%D0%BA_(%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F%D0%BB%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B)&action=edit&redlink=1" class="new" title="Коүҙәк (категориялар теорияһы) (был бит юҡ)">коүҙәктәр</a>) бар тигәнде аңлата. Берәмеге булған ҡулсалар категорияһының <a href="/w/index.php?title=%D0%91%D0%B0%D1%88%D0%BB%D0%B0%D0%BD%D2%93%D1%8B%D1%81_%D0%BE%D0%B1%D1%8A%D0%B5%D0%BA%D1%82&action=edit&redlink=1" class="new" title="Башланғыс объект (был бит юҡ)">башланғыс объекты</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> ҡулсаһы) һәм <a href="/w/index.php?title=%D0%A2%D0%B5%D1%80%D0%BC%D0%B8%D0%BD%D0%B0%D0%BB%D1%8C_%D0%BE%D0%B1%D1%8A%D0%B5%D0%BA%D1%82%D1%8B&action=edit&redlink=1" class="new" title="Терминаль объекты (был бит юҡ)">терминаль объекты</a> (нуль ҡулса) бар. </p><p>Ҡулсаға ошондай категориялы билдәләмә бирергә мөмкин: берәмеге булған ассоциатив ҡулса — <a href="/w/index.php?title=%D0%90%D0%B1%D0%B5%D0%BB%D1%8C_%D1%82%D3%A9%D1%80%D0%BA%D3%A9%D0%BC%D0%B4%D3%99%D1%80%D0%B5_%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Абель төркөмдәре категорияһы (был бит юҡ)">Абель төркөмдәре категорияһында</a> <a href="/w/index.php?title=%D0%9C%D0%BE%D0%BD%D0%BE%D0%B8%D0%B4_(%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F%D0%BB%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B)&action=edit&redlink=1" class="new" title="Моноид (категориялар теорияһы) (был бит юҡ)">моноид</a> ул (Абель төркөмдәре <a href="/w/index.php?title=%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80%D0%BB%D1%8B_%D2%A1%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D1%83&action=edit&redlink=1" class="new" title="Тензорлы ҡабатлау (был бит юҡ)">тензорлы ҡабатлау</a> операцияһына ҡарата <a href="/w/index.php?title=%D0%9C%D0%BE%D0%BD%D0%BE%D0%B8%D0%B4%D0%B0%D0%BB%D1%8C_%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Моноидаль категория (был бит юҡ)">моноидаль категория</a> барлыҡҡа килтерәләр). <i>R</i> ҡулсаһының Абель төркөмөндә (ҡабатлау буйынса <a href="/w/index.php?title=%D0%9C%D0%BE%D0%BD%D0%BE%D0%B8%D0%B4&action=edit&redlink=1" class="new" title="Моноид (был бит юҡ)">моноид</a> итеп ҡаралған ҡулсаның) <a href="/w/index.php?title=%D0%A2%D3%A9%D1%80%D0%BA%D3%A9%D0%BC_%D2%93%D3%99%D0%BC%D3%99%D0%BB%D0%B5&action=edit&redlink=1" class="new" title="Төркөм ғәмәле (был бит юҡ)">ғәмәл</a> Абель төркөмөн <i>R</i>-<a href="/wiki/%D2%A0%D1%83%D0%BB%D1%81%D0%B0_%D3%A9%D2%AB%D1%82%D3%A9%D0%BD%D0%B4%D3%99_%D0%BC%D0%BE%D0%B4%D1%83%D0%BB%D1%8C" title="Ҡулса өҫтөндә модуль">модулгә</a> әйләндерә. Модуль төшөнсәһе <a href="/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BB%D1%8B_%D0%B0%D1%80%D0%B0%D1%83%D1%8B%D2%A1" title="Векторлы арауыҡ">векторлы арауыҡ</a> төшөнсәһен дөйөмләштерә: тупаҫыраҡ итеп әйткәндә, модуль — «ҡулса өҫтөндө векторлы арауыҡ» ул.<sup id="cite_ref-Фейс—1977——_29-1" class="reference"><a href="#cite_note-Фейс—1977——-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Фейс—1979——_30-1" class="reference"><a href="#cite_note-Фейс—1979——-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Ҡулсаларҙың_махсус_кластары"><span id=".D2.A0.D1.83.D0.BB.D1.81.D0.B0.D0.BB.D0.B0.D1.80.D2.99.D1.8B.D2.A3_.D0.BC.D0.B0.D1.85.D1.81.D1.83.D1.81_.D0.BA.D0.BB.D0.B0.D1.81.D1.82.D0.B0.D1.80.D1.8B"></span>Ҡулсаларҙың махсус кластары</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=17" title="Ҡулсаларҙың махсус кластары бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=17" title="Ҡулсаларҙың махсус кластары бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2;"> <ul><li><a href="/w/index.php?title=%D0%90%D1%80%D1%82%D0%B8%D0%BD_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Артин ҡулсаһы (был бит юҡ)">Артин ҡулсаһы</a></li> <li><a href="/w/index.php?title=%D0%94%D0%B5%D0%B4%D0%B5%D0%BA%D0%B8%D0%BD%D0%B4_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Дедекинд ҡулсаһы (был бит юҡ)">Дедекинд ҡулсаһы</a></li> <li><a href="/w/index.php?title=%D0%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Дистрибутив ҡулса (был бит юҡ)">Дистрибутив ҡулса</a></li> <li><a href="/w/index.php?title=%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0&action=edit&redlink=1" class="new" title="Дифференциаль алгебра (был бит юҡ)">Дифференциаль ҡулса</a></li> <li><a href="/w/index.php?title=%D0%A2%D3%A9%D0%BF_%D0%B8%D0%B4%D0%B5%D0%B0%D0%BB%D0%B4%D0%B0%D1%80_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Төп идеалдар ҡулсаһы (был бит юҡ)">Төп идеалдар ҡулсаһы</a></li> <li><a href="/w/index.php?title=%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Евклид ҡулсаһы (был бит юҡ)">Евклид ҡулсаһы</a></li> <li><a href="/w/index.php?title=%D0%91%D0%B5%D0%B7%D1%83_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Безу ҡулсаһы (был бит юҡ)">Безу ҡулсаһы</a></li> <li><a href="/w/index.php?title=%D0%9B%D0%B8_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Ли ҡулсаһы (был бит юҡ)">Ли ҡулсаһы</a></li> <li><a href="/w/index.php?title=%D0%A1%D0%B8%D0%BA%D0%BB%D0%B5_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Сикле ҡулса (был бит юҡ)">Сикле ҡулса</a></li> <li><a href="/w/index.php?title=%D0%9B%D0%BE%D0%BA%D0%B0%D0%BB%D1%8C_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Локаль ҡулса (был бит юҡ)">Локаль ҡулса</a></li> <li><a href="/w/index.php?title=%D0%9D%D1%91%D1%82%D0%B5%D1%80_%D2%A1%D1%83%D0%BB%D1%81%D0%B0%D2%BB%D1%8B&action=edit&redlink=1" class="new" title="Нётер ҡулсаһы (был бит юҡ)">Нётер ҡулсаһы</a></li> <li><a href="/w/index.php?title=%D0%91%D3%A9%D1%82%D3%A9%D0%BD%D0%BB%D3%A9%D0%BA_%D3%A9%D0%BB%D0%BA%D3%99%D2%BB%D0%B5&action=edit&redlink=1" class="new" title="Бөтөнлөк өлкәһе (был бит юҡ)">Бөтөнлөк өлкәһе</a></li> <li><a href="/w/index.php?title=%D0%9E%D0%B1%D0%BB%D0%B0%D1%81%D1%82%D1%8C_%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D1%8B%D1%85_%D0%B8%D0%B4%D0%B5%D0%B0%D0%BB%D0%BE%D0%B2&action=edit&redlink=1" class="new" title="Область главных идеалов (был бит юҡ)">Область главных идеалов</a></li> <li><a href="/w/index.php?title=%D0%91%D0%B5%D1%80%D0%B5%D0%BD%D1%81%D0%B5%D0%BB_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Беренсел ҡулса (был бит юҡ)">Беренсел ҡулса</a></li> <li><a href="/w/index.php?title=%D0%AF%D1%80%D1%8B%D0%BC_%D0%BB%D0%BE%D0%BA%D0%B0%D0%BB%D1%8C_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Ярым локаль ҡулса (был бит юҡ)">Ярым локаль ҡулса</a></li> <li><a href="/w/index.php?title=%D0%AF%D1%80%D1%8B%D0%BC_%D0%B1%D0%B5%D1%80%D0%B5%D0%BD%D1%81%D0%B5%D0%BB_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Ярым беренсел ҡулса (был бит юҡ)">Ярым беренсел ҡулса</a></li> <li><a href="/w/index.php?title=%D0%AF%D1%80%D1%8B%D0%BC_%D1%8F%D0%B1%D0%B0%D0%B9_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Ярым ябай ҡулса (был бит юҡ)">Ярым ябай ҡулса</a></li> <li><a href="/w/index.php?title=%D0%AF%D1%80%D1%8B%D0%BC_%D1%81%D1%8B%D0%BD%D0%B9%D1%8B%D1%80%D0%BB%D1%8B_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Ярым сынйырлы ҡулса (был бит юҡ)">Ярым сынйырлы ҡулса</a></li> <li><a href="/w/index.php?title=%D0%AF%D0%B1%D0%B0%D0%B9_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Ябай ҡулса (был бит юҡ)">Ябай ҡулса</a></li> <li><a href="/w/index.php?title=%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B0%D0%BB%D1%8C_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Факториаль ҡулса (был бит юҡ)">Факториаль ҡулса</a></li> <li><a href="/w/index.php?title=%D0%A1%D1%8B%D0%BD%D0%B9%D1%8B%D1%80%D0%BB%D1%8B_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Сынйырлы ҡулса (был бит юҡ)">Сынйырлы ҡулса</a></li></ul> </div> <p>Дөйөмләштереүҙәр — <a href="/w/index.php?title=%D0%90%D1%81%D1%81%D0%BE%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2_%D0%B1%D1%83%D0%BB%D0%BC%D0%B0%D2%93%D0%B0%D0%BD_%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Ассоциатив булмаған ҡулса (был бит юҡ)">ассоциатив булмаған ҡулса</a>, <a href="/w/index.php?title=%D0%AF%D1%80%D1%8B%D0%BC%D2%A1%D1%83%D0%BB%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Ярымҡулса (был бит юҡ)">ярымҡулса</a>, <a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0%D2%93%D0%B0_%D1%8F%D2%A1%D1%8B%D0%BD&action=edit&redlink=1" class="new" title="Ҡулсаға яҡын (был бит юҡ)">ҡулсаға яҡын</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Ҡулсалар_өҫтөндә_структуралар"><span id=".D2.A0.D1.83.D0.BB.D1.81.D0.B0.D0.BB.D0.B0.D1.80_.D3.A9.D2.AB.D1.82.D3.A9.D0.BD.D0.B4.D3.99_.D1.81.D1.82.D1.80.D1.83.D0.BA.D1.82.D1.83.D1.80.D0.B0.D0.BB.D0.B0.D1.80"></span>Ҡулсалар өҫтөндә структуралар</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=18" title="Ҡулсалар өҫтөндә структуралар бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=18" title="Ҡулсалар өҫтөндә структуралар бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_%D3%A9%D2%AB%D1%82%D3%A9%D0%BD%D0%B4%D3%99_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0&action=edit&redlink=1" class="new" title="Ҡулса өҫтөндә алгебра (был бит юҡ)">Ҡулса өҫтөндә алгебра</a></li> <li><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_%D3%A9%D2%AB%D1%82%D3%A9%D0%BD%D0%B4%D3%99_%D0%B1%D0%B8%D0%BC%D0%BE%D0%B4%D1%83%D0%BB%D1%8C&action=edit&redlink=1" class="new" title="Ҡулса өҫтөндә бимодуль (был бит юҡ)">Ҡулса өҫтөндә бимодуль</a></li> <li><a href="/wiki/%D2%A0%D1%83%D0%BB%D1%81%D0%B0_%D3%A9%D2%AB%D1%82%D3%A9%D0%BD%D0%B4%D3%99_%D0%BC%D0%BE%D0%B4%D1%83%D0%BB%D1%8C" title="Ҡулса өҫтөндә модуль">Ҡулса өҫтөндә модуль</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Иҫкәрмәләр"><span id=".D0.98.D2.AB.D0.BA.D3.99.D1.80.D0.BC.D3.99.D0.BB.D3.99.D1.80"></span>Иҫкәрмәләр</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=19" title="Иҫкәрмәләр бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=19" title="Иҫкәрмәләр бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small" style=""> <ol class="references"> <li id="cite_note-Винберг—2011——17—19-1"><span class="mw-cite-backlink"><a href="#cite_ref-Винберг—2011——17—19_1-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 17—19</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="citation"><i>Бельский А., Садовский Л.</i> <a rel="nofollow" class="external text" href="http://kvant.mccme.ru/1974/02/kolca.htm">Кольца</a> // <i><a href="/w/index.php?title=%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_(%D0%B6%D1%83%D1%80%D0%BD%D0%B0%D0%BB)&action=edit&redlink=1" class="new" title="Квант (журнал) (был бит юҡ)">Квант</a></i>. — 1974. — № 2.</span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation"><i>Erich Reck</i> <a rel="nofollow" class="external text" href="http://plato.stanford.edu/archives/win2012/entries/dedekind-foundations/">Dedekind's Contributions to the Foundations of Mathematics</a> // <i>The Stanford Encyclopedia of Philosophy</i>  / Edward N. Zalta. — 2012-01-01.</span></span> </li> <li id="cite_note-Атья,_Макдональд—1972——9-4"><span class="mw-cite-backlink"><a href="#cite_ref-Атья,_Макдональд—1972——9_4-0">↑</a></span> <span class="reference-text"><a href="#CITEREFАтья,_Макдональд1972">Атья, Макдональд, 1972</a>, с. 9</span> </li> <li id="cite_note-Винберг—2011——18—19-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Винберг—2011——18—19_5-0">5,0</a></sup> <sup><a href="#cite_ref-Винберг—2011——18—19_5-1">5,1</a></sup> <sup><a href="#cite_ref-Винберг—2011——18—19_5-2">5,2</a></sup> <sup><a href="#cite_ref-Винберг—2011——18—19_5-3">5,3</a></sup></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 18—19</span> </li> <li id="cite_note-Курош—1968——273—275-6"><span class="mw-cite-backlink"><a href="#cite_ref-Курош—1968——273—275_6-0">↑</a></span> <span class="reference-text"><a href="#CITEREFКурош1968">Курош, 1968</a>, с. 273—275</span> </li> <li id="cite_note-Ван_дер_Варден—1975——51—53-7"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Ван_дер_Варден—1975——51—53_7-0">7,0</a></sup> <sup><a href="#cite_ref-Ван_дер_Варден—1975——51—53_7-1">7,1</a></sup></span> <span class="reference-text"><a href="#CITEREFВан_дер_Варден1975">Ван дер Варден, 1975</a>, с. 51—53</span> </li> <li id="cite_note-Атья,_Макдональд—1972——11-8"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Атья,_Макдональд—1972——11_8-0">8,0</a></sup> <sup><a href="#cite_ref-Атья,_Макдональд—1972——11_8-1">8,1</a></sup></span> <span class="reference-text"><a href="#CITEREFАтья,_Макдональд1972">Атья, Макдональд, 1972</a>, с. 11</span> </li> <li id="cite_note-Ван_дер_Варден—1975——359-9"><span class="mw-cite-backlink"><a href="#cite_ref-Ван_дер_Варден—1975——359_9-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВан_дер_Варден1975">Ван дер Варден, 1975</a>, с. 359</span> </li> <li id="cite_note-Винберг—2011——407-10"><span class="mw-cite-backlink"><a href="#cite_ref-Винберг—2011——407_10-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 407</span> </li> <li id="cite_note-Куликов—1979——110—111-11"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Куликов—1979——110—111_11-0">11,0</a></sup> <sup><a href="#cite_ref-Куликов—1979——110—111_11-1">11,1</a></sup> <sup><a href="#cite_ref-Куликов—1979——110—111_11-2">11,2</a></sup></span> <span class="reference-text"><a href="#CITEREFКуликов1979">Куликов, 1979</a>, с. 110—111</span> </li> <li id="cite_note-Винберг—2011——21-12"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Винберг—2011——21_12-0">12,0</a></sup> <sup><a href="#cite_ref-Винберг—2011——21_12-1">12,1</a></sup></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 21</span> </li> <li id="cite_note-Куликов—1979——437-13"><span class="mw-cite-backlink"><a href="#cite_ref-Куликов—1979——437_13-0">↑</a></span> <span class="reference-text"><a href="#CITEREFКуликов1979">Куликов, 1979</a>, с. 437</span> </li> <li id="cite_note-Ван_дер_Варден—1975——64-14"><span class="mw-cite-backlink"><a href="#cite_ref-Ван_дер_Варден—1975——64_14-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВан_дер_Варден1975">Ван дер Варден, 1975</a>, с. 64</span> </li> <li id="cite_note-Фейс—1977——153-15"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Фейс—1977——153_15-0">15,0</a></sup> <sup><a href="#cite_ref-Фейс—1977——153_15-1">15,1</a></sup></span> <span class="reference-text"><a href="#CITEREFФейс1977">Фейс, 1977</a>, с. 153</span> </li> <li id="cite_note-Куликов—1979——430—431-16"><span class="mw-cite-backlink"><a href="#cite_ref-Куликов—1979——430—431_16-0">↑</a></span> <span class="reference-text"><a href="#CITEREFКуликов1979">Куликов, 1979</a>, с. 430—431</span> </li> <li id="cite_note-Винберг—2011——406-17"><span class="mw-cite-backlink"><a href="#cite_ref-Винберг—2011——406_17-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 406</span> </li> <li id="cite_note-Фейс—1979——10-18"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Фейс—1979——10_18-0">18,0</a></sup> <sup><a href="#cite_ref-Фейс—1979——10_18-1">18,1</a></sup></span> <span class="reference-text"><a href="#CITEREFФейс1979">Фейс, 1979</a>, с. 10</span> </li> <li id="cite_note-Винберг—2011——388-19"><span class="mw-cite-backlink"><a href="#cite_ref-Винберг—2011——388_19-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 388</span> </li> <li id="cite_note-Куликов—1979——107—108-20"><span class="mw-cite-backlink"><a href="#cite_ref-Куликов—1979——107—108_20-0">↑</a></span> <span class="reference-text"><a href="#CITEREFКуликов1979">Куликов, 1979</a>, с. 107—108</span> </li> <li id="cite_note-Куликов—1979——432-21"><span class="mw-cite-backlink"><a href="#cite_ref-Куликов—1979——432_21-0">↑</a></span> <span class="reference-text"><a href="#CITEREFКуликов1979">Куликов, 1979</a>, с. 432</span> </li> <li id="cite_note-Винберг—2011——387—390-22"><span class="mw-cite-backlink"><a href="#cite_ref-Винберг—2011——387—390_22-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 387—390</span> </li> <li id="cite_note-Винберг—2011——523-23"><span class="mw-cite-backlink"><a href="#cite_ref-Винберг—2011——523_23-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 523</span> </li> <li id="cite_note-Фейс—1977——152-24"><span class="mw-cite-backlink"><a href="#cite_ref-Фейс—1977——152_24-0">↑</a></span> <span class="reference-text"><a href="#CITEREFФейс1977">Фейс, 1977</a>, с. 152</span> </li> <li id="cite_note-Куликов—1979——430-25"><span class="mw-cite-backlink"><a href="#cite_ref-Куликов—1979——430_25-0">↑</a></span> <span class="reference-text"><a href="#CITEREFКуликов1979">Куликов, 1979</a>, с. 430</span> </li> <li id="cite_note-Винберг—2011——118-26"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Винберг—2011——118_26-0">26,0</a></sup> <sup><a href="#cite_ref-Винберг—2011——118_26-1">26,1</a></sup></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 118</span> </li> <li id="cite_note-Атья,_Макдональд—1972——-27"><span class="mw-cite-backlink"><a href="#cite_ref-Атья,_Макдональд—1972——_27-0">↑</a></span> <span class="reference-text"><a href="#CITEREFАтья,_Макдональд1972">Атья, Макдональд, 1972</a></span> </li> <li id="cite_note-Курош—1968——266-28"><span class="mw-cite-backlink"><a href="#cite_ref-Курош—1968——266_28-0">↑</a></span> <span class="reference-text"><a href="#CITEREFКурош1968">Курош, 1968</a>, с. 266</span> </li> <li id="cite_note-Фейс—1977——-29"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Фейс—1977——_29-0">29,0</a></sup> <sup><a href="#cite_ref-Фейс—1977——_29-1">29,1</a></sup></span> <span class="reference-text"><a href="#CITEREFФейс1977">Фейс, 1977</a></span> </li> <li id="cite_note-Фейс—1979——-30"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Фейс—1979——_30-0">30,0</a></sup> <sup><a href="#cite_ref-Фейс—1979——_30-1">30,1</a></sup></span> <span class="reference-text"><a href="#CITEREFФейс1979">Фейс, 1979</a></span> </li> <li id="cite_note-Винберг—2011——28—34-31"><span class="mw-cite-backlink"><a href="#cite_ref-Винберг—2011——28—34_31-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВинберг2011">Винберг, 2011</a>, с. 28—34</span> </li> <li id="cite_note-Ван_дер_Варден—1975——509—512-32"><span class="mw-cite-backlink"><a href="#cite_ref-Ван_дер_Варден—1975——509—512_32-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВан_дер_Варден1975">Ван дер Варден, 1975</a>, с. 509—512</span> </li> <li id="cite_note-Ван_дер_Варден—1975——33-33"><span class="mw-cite-backlink"><a href="#cite_ref-Ван_дер_Варден—1975——33_33-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВан_дер_Варден1975">Ван дер Варден, 1975</a>, с. 33</span> </li> <li id="cite_note-Ван_дер_Варден—1975——173-34"><span class="mw-cite-backlink"><a href="#cite_ref-Ван_дер_Варден—1975——173_34-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВан_дер_Варден1975">Ван дер Варден, 1975</a>, с. 173</span> </li> <li id="cite_note-Ван_дер_Варден—1975——450—452-35"><span class="mw-cite-backlink"><a href="#cite_ref-Ван_дер_Варден—1975——450—452_35-0">↑</a></span> <span class="reference-text"><a href="#CITEREFВан_дер_Варден1975">Ван дер Варден, 1975</a>, с. 450—452</span> </li> <li id="cite_note-Курош—1968——305—311-36"><span class="mw-cite-backlink"><a href="#cite_ref-Курош—1968——305—311_36-0">↑</a></span> <span class="reference-text"><a href="#CITEREFКурош1968">Курош, 1968</a>, с. 305—311</span> </li> </ol> </div> <div class="mw-heading mw-heading2"><h2 id="Әҙәбиәт"><span id=".D3.98.D2.99.D3.99.D0.B1.D0.B8.D3.99.D1.82"></span>Әҙәбиәт</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&veaction=edit&section=20" title="Әҙәбиәт бүлеген мөхәррирләргә" class="mw-editsection-visualeditor"><span>үҙгәртергә</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&section=20" title="Әҙәбиәт бүлегенең сығанаҡ кодын мөхәррирләү"><span>сығанаҡты үҙгәртеү</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFАтья,_Макдональд1972"><i><a href="/w/index.php?title=%D0%90%D1%82%D1%8C%D1%8F,_%D0%9C%D0%B0%D0%B9%D0%BA%D0%BB_%D0%A4%D1%80%D1%8D%D0%BD%D1%81%D0%B8%D1%81&action=edit&redlink=1" class="new" title="Атья, Майкл Фрэнсис (был бит юҡ)">М. Атья</a>, И. Макдональд.</i> Введение в коммутативную алгебру. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: Мир, 1972. — 160 с.</span></li> <li><i>Бельский А., Садовский Л.</i> <a rel="nofollow" class="external text" href="http://kvant.mccme.ru/1974/02/kolca.htm">Кольца.</a> <a href="/w/index.php?title=%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_(%D0%B6%D1%83%D1%80%D0%BD%D0%B0%D0%BB)&action=edit&redlink=1" class="new" title="Квант (журнал) (был бит юҡ)">Квант</a> № 2, 1974.</li> <li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFВан_дер_Варден1975"><i><a href="/w/index.php?title=%D0%92%D0%B0%D0%BD_%D0%B4%D0%B5%D1%80_%D0%92%D0%B0%D1%80%D0%B4%D0%B5%D0%BD,_%D0%91%D0%B0%D1%80%D1%82%D0%B5%D0%BB%D1%8C_%D0%9B%D0%B5%D0%B5%D0%BD%D0%B4%D0%B5%D1%80%D1%82&action=edit&redlink=1" class="new" title="Ван дер Варден, Бартель Леендерт (был бит юҡ)">Ван дер Варден Б. Л.</a></i> Алгебра. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: Мир, 1975. — 623 с.</span></li> <li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFВинберг2011"><i><a href="/w/index.php?title=%D0%92%D0%B8%D0%BD%D0%B1%D0%B5%D1%80%D0%B3,_%D0%AD%D1%80%D0%BD%D0%B5%D1%81%D1%82_%D0%91%D0%BE%D1%80%D0%B8%D1%81%D0%BE%D0%B2%D0%B8%D1%87&action=edit&redlink=1" class="new" title="Винберг, Эрнест Борисович (был бит юҡ)">Винберг Э. Б.</a></i> Курс алгебры. - Новое издание, перераб. и доп.. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: МЦНМО, 2011. — 592 с.</span></li> <li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFГлейзер1983"><i><a href="/w/index.php?title=%D0%93%D0%BB%D0%B5%D0%B9%D0%B7%D0%B5%D1%80,_%D0%93%D0%B5%D1%80%D1%88_%D0%98%D1%81%D0%B0%D0%B0%D0%BA%D0%BE%D0%B2%D0%B8%D1%87&action=edit&redlink=1" class="new" title="Глейзер, Герш Исаакович (был бит юҡ)">Глейзер Г. И.</a></i> История математики в школе: IX-X класс. Пособие для учителей - Новое издание, перераб. и доп.. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: Просвещение, 1983. — 351 с.</span></li> <li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFКолмогоров1978"><i><a href="/w/index.php?title=%D0%9A%D0%BE%D0%BB%D0%BC%D0%BE%D0%B3%D0%BE%D1%80%D0%BE%D0%B2,_%D0%90%D0%BD%D0%B4%D1%80%D0%B5%D0%B9_%D0%9D%D0%B8%D0%BA%D0%BE%D0%BB%D0%B0%D0%B5%D0%B2%D0%B8%D1%87&action=edit&redlink=1" class="new" title="Колмогоров, Андрей Николаевич (был бит юҡ)">Колмогоров А. Н.</a>,<a href="/w/index.php?title=%D0%AE%D1%88%D0%BA%D0%B5%D0%B2%D0%B8%D1%87,_%D0%90%D0%B4%D0%BE%D0%BB%D1%8C%D1%84_%D0%9F%D0%B0%D0%B2%D0%BB%D0%BE%D0%B2%D0%B8%D1%87&action=edit&redlink=1" class="new" title="Юшкевич, Адольф Павлович (был бит юҡ)">Юшкевич А. П.</a>(ред.).</i> Математика XIX века. Математическая логика. Алгебра. Теория чисел. Теория вероятностей. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: Наука, 1978. — 255 с.</span></li> <li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFКуликов1979"><i>Куликов Л. Я.</i> Алгебра и теория чисел: Учеб. пособие для педагогических институтов. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: Высш. школа, 1979. — 559 с.</span></li> <li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFКурош1968"><i><a href="/w/index.php?title=%D0%9A%D1%83%D1%80%D0%BE%D1%88,_%D0%90%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD%D0%B4%D1%80_%D0%93%D0%B5%D0%BD%D0%BD%D0%B0%D0%B4%D0%B8%D0%B5%D0%B2%D0%B8%D1%87&action=edit&redlink=1" class="new" title="Курош, Александр Геннадиевич (был бит юҡ)">Курош А. Г.</a></i> Курс высшей алгебры.. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: Наука, 1968. — 431 с.</span></li> <li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFФейс1977"><i>Фейс К.</i> Алгебра. Кольца, модули, категории.. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: Мир, 1977. — Т. 1. — 688 с.</span></li> <li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFФейс1979"><i>Фейс К.</i> Алгебра. Кольца, модули, категории.. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: Мир, 1979. — Т. 2. — 464 с.</span></li> <li><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFХерстейн1972"><i>Херстейн И.</i> Некоммутативные кольца. — <span style="border-bottom:1px dotted gray; cursor:default" title="Москва">М</span>.: Мир, 1972. — 190 с.</span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">«<a dir="ltr" href="https://ba.wikipedia.org/w/index.php?title=Ҡулса_(математика)&oldid=864472">https://ba.wikipedia.org/w/index.php?title=Ҡулса_(математика)&oldid=864472</a>» битенән алынған</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F%D0%BB%D0%B0%D1%80" title="Махсус:Категориялар">Категория</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D2%A0%D1%83%D0%BB%D1%81%D0%B0%D0%BB%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Категория:Ҡулсалар теорияһы">Ҡулсалар теорияһы</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Йәшерелгән категория: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%91%D0%B0%D1%88%D2%A1%D0%B0_%D0%BC%D3%99%D2%93%D3%99%D0%BD%D3%99%D0%BB%D3%99%D1%80%D0%B5:_%D0%B1%D1%83%D0%BB%D0%BC%D0%B0%D2%93%D0%B0%D0%BD_%D0%BC%D3%99%D2%A1%D3%99%D0%BB%D3%99_%D0%BA%D2%AF%D1%80%D2%BB%D3%99%D1%82%D0%B5%D0%BB%D0%B3%D3%99%D0%BD" title="Категория:Башҡа мәғәнәләре: булмаған мәҡәлә күрһәтелгән">Башҡа мәғәнәләре: булмаған мәҡәлә күрһәтелгән</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Был бит һуңғы тапҡыр 09:07 7 февраль 2019 үҙгәртелгән.</li> <li id="footer-info-copyright">Текст <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike License</a> буйынса рөхсәт ителгән; өҫтәмә шарттар ҡулланылыуы мөмкин. 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