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class="vector-toc-text"> <span class="vector-toc-numb">一</span> <span>定義</span> </div> </a> <ul id="toc-定義-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-例" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例"> <div class="vector-toc-text"> <span class="vector-toc-numb">二</span> <span>例</span> </div> </a> <ul id="toc-例-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-註" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#註"> <div class="vector-toc-text"> <span class="vector-toc-numb">三</span> <span>註</span> </div> </a> <ul id="toc-註-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="目次" 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="切換目次" > <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">切換目次</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">環 (代數)</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="前往別種語言的條目。有六六種語言可用" > <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">六六種語言</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) – 阿拉伯文" lang="ar" hreflang="ar" data-title="حلقة (رياضيات)" data-language-autonym="العربية" data-language-local-name="阿拉伯文" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%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)" title="Ҡулса (математика) – 巴什喀爾文" lang="ba" hreflang="ba" data-title="Ҡулса (математика)" data-language-autonym="Башҡортса" data-language-local-name="巴什喀爾文" class="interlanguage-link-target"><span>Башҡортса</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) – 白俄羅斯文" lang="be" hreflang="be" data-title="Кальцо (алгебра)" data-language-autonym="Беларуская" data-language-local-name="白俄羅斯文" 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) – 保加利亞文" lang="bg" hreflang="bg" data-title="Пръстен (алгебра)" data-language-autonym="Български" data-language-local-name="保加利亞文" 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) – 波士尼亞文" lang="bs" hreflang="bs" data-title="Prsten (matematika)" data-language-autonym="Bosanski" data-language-local-name="波士尼亞文" 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) – 加泰蘭文" lang="ca" hreflang="ca" data-title="Anell (matemàtiques)" data-language-autonym="Català" data-language-local-name="加泰蘭文" 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) – 捷克文" lang="cs" hreflang="cs" data-title="Okruh (algebra)" data-language-autonym="Čeština" data-language-local-name="捷克文" 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) – 楚瓦什文" lang="cv" hreflang="cv" data-title="Ункă (математика)" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什文" 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) – 威爾斯文" lang="cy" hreflang="cy" data-title="Modrwy (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="威爾斯文" 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) – 丹麥文" lang="da" hreflang="da" data-title="Ring (matematik)" data-language-autonym="Dansk" data-language-local-name="丹麥文" 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) – 德文" lang="de" hreflang="de" data-title="Ring (Algebra)" data-language-autonym="Deutsch" data-language-local-name="德文" 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) – 希臘文" lang="el" hreflang="el" data-title="Δακτύλιος (άλγεβρα)" data-language-autonym="Ελληνικά" data-language-local-name="希臘文" 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) – 英文" lang="en" hreflang="en" data-title="Ring (mathematics)" data-language-autonym="English" data-language-local-name="英文" 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) – 世界文" lang="eo" hreflang="eo" data-title="Ringo (algebro)" data-language-autonym="Esperanto" data-language-local-name="世界文" 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) – 西班牙文" lang="es" hreflang="es" data-title="Anillo (matemática)" data-language-autonym="Español" data-language-local-name="西班牙文" 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) – 愛沙尼亞文" lang="et" hreflang="et" data-title="Ring (algebra)" data-language-autonym="Eesti" data-language-local-name="愛沙尼亞文" 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) – 巴斯克文" lang="eu" hreflang="eu" data-title="Eraztun (matematika)" data-language-autonym="Euskara" data-language-local-name="巴斯克文" 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) – 波斯文" lang="fa" hreflang="fa" data-title="حلقه (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="波斯文" 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) – 芬蘭文" lang="fi" hreflang="fi" data-title="Rengas (matematiikka)" data-language-autonym="Suomi" data-language-local-name="芬蘭文" 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) – 法文" lang="fr" hreflang="fr" data-title="Anneau (mathématiques)" data-language-autonym="Français" data-language-local-name="法文" 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) – 愛爾蘭文" lang="ga" hreflang="ga" data-title="Fáinne (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="愛爾蘭文" 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) – 加利西亞文" lang="gl" hreflang="gl" data-title="Anel (álxebra)" data-language-autonym="Galego" data-language-local-name="加利西亞文" 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) – 希伯來文" lang="he" hreflang="he" data-title="חוג (מבנה אלגברי)" data-language-autonym="עברית" data-language-local-name="希伯來文" 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) – 克羅埃西亞文" lang="hr" hreflang="hr" data-title="Prsten (matematika)" data-language-autonym="Hrvatski" data-language-local-name="克羅埃西亞文" 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) – 匈牙利文" lang="hu" hreflang="hu" data-title="Gyűrű (matematika)" data-language-autonym="Magyar" data-language-local-name="匈牙利文" 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) – 亞美尼亞文" lang="hy" hreflang="hy" data-title="Օղակ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="亞美尼亞文" 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) – 國際文" lang="ia" hreflang="ia" data-title="Anello (algebra)" data-language-autonym="Interlingua" data-language-local-name="國際文" 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) – 印尼文" lang="id" hreflang="id" data-title="Gelanggang (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="印尼文" 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) – 義大利文" lang="it" hreflang="it" data-title="Anello (algebra)" data-language-autonym="Italiano" data-language-local-name="義大利文" 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) – 日文" lang="ja" hreflang="ja" data-title="環 (数学)" data-language-autonym="日本語" data-language-local-name="日文" 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) – 喬治亞文" lang="ka" hreflang="ka" data-title="რგოლი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="喬治亞文" 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 – 坎那達文" lang="kn" hreflang="kn" data-title="ರಿಂಗ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="坎那達文" 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) – 韓文" lang="ko" hreflang="ko" data-title="환 (수학)" data-language-autonym="한국어" data-language-local-name="韓文" 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 – 拉丁文" lang="la" hreflang="la" data-title="Anellus" data-language-autonym="Latina" data-language-local-name="拉丁文" 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) – 盧森堡文" lang="lb" hreflang="lb" data-title="Rank (Algeber)" data-language-autonym="Lëtzebuergesch" data-language-local-name="盧森堡文" 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) – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Anell (matematega)" data-language-autonym="Lombard" data-language-local-name="倫巴底文" 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) – 馬來亞拉姆文" lang="ml" hreflang="ml" data-title="വലയം (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="馬來亞拉姆文" 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 – 馬拉地文" lang="mr" hreflang="mr" data-title="रिंग" data-language-autonym="मराठी" data-language-local-name="馬拉地文" 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) – 馬來文" lang="ms" hreflang="ms" data-title="Gelanggang (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="馬來文" 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) – 荷蘭文" lang="nl" hreflang="nl" data-title="Ring (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="荷蘭文" 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 – 耐諾斯克挪威文" lang="nn" hreflang="nn" data-title="Ring i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="耐諾斯克挪威文" 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) – 巴克摩挪威文" lang="nb" hreflang="nb" data-title="Ring (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="巴克摩挪威文" 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) – 諾維亞文" lang="nov" hreflang="nov" data-title="Ringe (matematike)" data-language-autonym="Novial" data-language-local-name="諾維亞文" 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) – 波蘭文" lang="pl" hreflang="pl" data-title="Pierścień (matematyka)" data-language-autonym="Polski" data-language-local-name="波蘭文" 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 – 皮埃蒙特文" lang="pms" hreflang="pms" data-title="Anel" data-language-autonym="Piemontèis" data-language-local-name="皮埃蒙特文" 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) – 葡萄牙文" lang="pt" hreflang="pt" data-title="Anel (matemática)" data-language-autonym="Português" data-language-local-name="葡萄牙文" 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) – 羅馬尼亞文" lang="ro" hreflang="ro" data-title="Inel (matematică)" data-language-autonym="Română" data-language-local-name="羅馬尼亞文" 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) – 俄文" lang="ru" hreflang="ru" data-title="Кольцо (математика)" data-language-autonym="Русский" data-language-local-name="俄文" 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) – 西西里文" lang="scn" hreflang="scn" data-title="Aneddu (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="西西里文" 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 – 塞爾維亞克羅埃西亞文" lang="sh" hreflang="sh" data-title="Algebarski prsten" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞爾維亞克羅埃西亞文" 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) – 斯洛伐克文" lang="sk" hreflang="sk" data-title="Okruh (algebra)" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克文" 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) – 斯洛維尼亞文" lang="sl" hreflang="sl" data-title="Kolobar (algebra)" data-language-autonym="Slovenščina" data-language-local-name="斯洛維尼亞文" 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 – 塞爾維亞文" lang="sr" hreflang="sr" data-title="Алгебарски прстен" data-language-autonym="Српски / srpski" data-language-local-name="塞爾維亞文" 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) – 瑞典文" lang="sv" hreflang="sv" data-title="Ring (matematik)" data-language-autonym="Svenska" data-language-local-name="瑞典文" 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) – 坦米爾文" lang="ta" hreflang="ta" data-title="வளையம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="坦米爾文" 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) – 泰文" lang="th" hreflang="th" data-title="ริง (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="泰文" 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 – 土耳其文" lang="tr" hreflang="tr" data-title="Halka" data-language-autonym="Türkçe" data-language-local-name="土耳其文" 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) – 烏克蘭文" lang="uk" hreflang="uk" data-title="Кільце (алгебра)" data-language-autonym="Українська" data-language-local-name="烏克蘭文" 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) – 烏都文" lang="ur" hreflang="ur" data-title="حلقہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="烏都文" 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 – 越南文" lang="vi" hreflang="vi" data-title="Vành" data-language-autonym="Tiếng Việt" data-language-local-name="越南文" 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) – 西佛蘭德文" lang="vls" hreflang="vls" data-title="Rienk (algebra)" data-language-autonym="West-Vlams" data-language-local-name="西佛蘭德文" 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） – 吳語" lang="wuu" hreflang="wuu" data-title="环（代数）" data-language-autonym="吴语" data-language-local-name="吳語" 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) – 中文" lang="zh" hreflang="zh" data-title="环 (代数)" data-language-autonym="中文" data-language-local-name="中文" class="interlanguage-link-target"><span>中文</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 – 閩南語" lang="nan" hreflang="nan" data-title="Khôan" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="閩南語" 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id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div id="mw-fr-revision-details" class="mw-fr-revision-details-dialog" style="display:none;"><div tabindex="0"></div><div class="cdx-dialog cdx-dialog--horizontal-actions"><header class="cdx-dialog__header cdx-dialog__header--default"><div class="cdx-dialog__header__title-group"><h2 class="cdx-dialog__header__title">頁面版本狀態</h2><p class="cdx-dialog__header__subtitle">該頁面尚未檢查</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
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							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">此為底本，未經<a href="/wiki/%E8%AA%AA%E6%98%8E:%E6%A0%A1%E5%B0%8D" title="說明:校對">審校</a>。</div></div><div tabindex="0"></div></div><div class="mw-content-ltr mw-parser-output" lang="lzh" dir="ltr"><div style="border:1px solid #A3B1BF; padding:0.5em 2em; width:60%; float:left;"> 註︰蓋當今數學之事，誠難僅以文述，而無符號，故凡數學之文，咸有漢字、拉丁字相易之事，以合文言、數學，則無論文理之人，皆可明之也。</div> <div style="clear:left;"></div> <p><b>環</b>者，有加乘二法之<a href="/wiki/%E4%BB%A3%E6%95%B8" class="mw-redirect" title="代數">代數</a>結構也。 </p> <div class="mw-heading mw-heading2"><h2 id="定義"><span id=".E5.AE.9A.E7.BE.A9"></span>定義</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%92%B0_(%E4%BB%A3%E6%95%B8)&action=edit&section=1" title="所纂章名：定義"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>環</b>者，集也，有<a href="/wiki/%E4%BA%8C%E5%85%83%E9%81%8B%E7%AE%97" title="二元運算">加乘</a>二法（「+ ，×」），合： </p> <ul><li>集合加者，<a href="/wiki/%E4%BA%A4%E6%8F%9B%E7%BE%A4" title="交換群">交換群</a>也，有單位元曰「零」。</li> <li>集合乘者，<a href="/w/index.php?title=%E5%8D%8A%E7%BE%A4&action=edit&redlink=1" class="new" title="半群（查無此頁）">半群</a>也，即守<a href="/w/index.php?title=%E5%B0%81%E9%96%89%E6%80%A7&action=edit&redlink=1" class="new" title="封閉性（查無此頁）">封閉性</a>與<a href="/wiki/%E7%B5%90%E5%90%88%E5%BE%8B" class="mw-redirect" title="結合律">結合律</a>。 <ul><li>甲乙之積乘丙，同乎甲乘乙丙之積也（「(x × y) × z = x × ( y × z)」），曰<a href="/wiki/%E7%B5%90%E5%90%88%E5%BE%8B" class="mw-redirect" title="結合律">結合律</a>。</li></ul></li> <li>甲乘乙丙之和，同乎甲乙之積加甲丙之積；甲乙之和乘丙，同乎甲丙之積加乙丙之積，曰<a href="/wiki/%E5%88%86%E9%85%8D%E5%BE%8B" class="mw-redirect" title="分配律">分配律</a>。（「x × ( y + z ) = x × y + x × z; (x + y) × z = x × z + y × z」<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>一<span class="cite-bracket">]</span></a></sup>）</li></ul> <p>可得證，物乘「零」同乎「零」乘物，皆「零」也（「x × 0 = 0 × x = 0」）。 </p> <ul><li>今所究者，多有乘法單位元曰「一」，且凡物乘「一」或物乘以「一」，皆為己（「1 o x = x o 1 = x」）。</li></ul> <p>若乘法合<a href="/wiki/%E4%BA%A4%E6%8F%9B%E5%BE%8B" class="mw-redirect" title="交換律">交換律</a>者，曰<b>交換環</b>。交換環而非零物皆有乘法逆者，曰<a href="/wiki/%E5%9F%9F_(%E4%BB%A3%E6%95%B8)" title="域 (代數)">域</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="例"><span id=".E4.BE.8B"></span>例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%92%B0_(%E4%BB%A3%E6%95%B8)&action=edit&section=2" title="所纂章名：例"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%95%B4%E6%95%B8" title="整數">整數</a>合加乘，交換環也。</li> <li><a href="/wiki/%E7%9F%A9%E9%99%A3" title="矩陣">方陣</a>合加乘，環也。然非交換環耳。</li> <li><a href="/wiki/%E5%9B%9B%E5%85%83%E6%95%B8" title="四元數">四元數</a>，環也。然非交換環耳。</li> <li><a href="/w/index.php?title=%E5%90%8C%E9%A4%98&action=edit&redlink=1" class="new" title="同餘（查無此頁）">同餘</a>集，交換環也。</li> <li><a href="/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式">多項式</a>集，交換環也。</li></ul> <div class="mw-heading mw-heading2"><h2 id="註"><span id=".E8.A8.BB"></span>註</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%92%B0_(%E4%BB%A3%E6%95%B8)&action=edit&section=3" title="所纂章名：註"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">依習，先乘除後加減。</span> </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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