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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-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">結合代數</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。有一七種語言可用" > <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-17" 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">一七種語言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_associativa" title="Àlgebra associativa – 加泰蘭文" lang="ca" hreflang="ca" data-title="Àlgebra associativa" 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-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Assoziative_Algebra" title="Assoziative Algebra – 德文" lang="de" hreflang="de" data-title="Assoziative 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-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Associative_algebra" title="Associative algebra – 英文" lang="en" hreflang="en" data-title="Associative algebra" 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/Asocieca_al%C4%9Debro" title="Asocieca alĝebro – 世界文" lang="eo" hreflang="eo" data-title="Asocieca alĝebro" 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/%C3%81lgebra_asociativa" title="Álgebra asociativa – 西班牙文" lang="es" hreflang="es" data-title="Álgebra asociativa" 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-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%B4%D8%B1%DA%A9%D8%AA%E2%80%8C%D9%BE%D8%B0%DB%8C%D8%B1" title="جبر شرکت‌پذیر – 波斯文" lang="fa" hreflang="fa" data-title="جبر شرکت‌پذیر" 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-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_associative" title="Algèbre associative – 法文" lang="fr" hreflang="fr" data-title="Algèbre associative" data-language-autonym="Français" data-language-local-name="法文" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="אלגברה (מבנה אלגברי) – 希伯來文" lang="he" hreflang="he" data-title="אלגברה (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/Asocijativna_algebra" title="Asocijativna algebra – 克羅埃西亞文" lang="hr" hreflang="hr" data-title="Asocijativna algebra" 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Algebra_associative" title="Algebra associative – 國際文" lang="ia" hreflang="ia" data-title="Algebra associative" 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/Aljabar_asosiatif" title="Aljabar asosiatif – 印尼文" lang="id" hreflang="id" data-title="Aljabar asosiatif" 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-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B5%90%E5%90%88%E5%A4%9A%E5%85%83%E7%92%B0" title="結合多元環 – 日文" lang="ja" hreflang="ja" data-title="結合多元環" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%B0%ED%95%A9_%EB%8C%80%EC%88%98" title="결합 대수 – 韓文" lang="ko" hreflang="ko" data-title="결합 대수" data-language-autonym="한국어" data-language-local-name="韓文" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Associatieve_algebra" title="Associatieve algebra – 荷蘭文" lang="nl" hreflang="nl" data-title="Associatieve algebra" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_associativa" title="Álgebra associativa – 葡萄牙文" lang="pt" hreflang="pt" data-title="Álgebra associativa" 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-uk badge-Q70893996 mw-list-item" title=""><a href="https://uk.wikipedia.org/wiki/%D0%90%D1%81%D0%BE%D1%86%D1%96%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Асоціативна алгебра – 烏克蘭文" lang="uk" hreflang="uk" data-title="Асоціативна 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%B5%90%E5%90%88%E4%BB%A3%E6%95%B8" title="結合代數 – 中文" lang="zh" hreflang="zh" data-title="結合代數" data-language-autonym="中文" data-language-local-name="中文" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q744960#sitelinks-wikipedia" title="修跨語鏈" class="wbc-editpage">纂鏈</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="名集"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet 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href="/wiki/%E7%89%B9%E6%AE%8A:%E6%8F%B4%E5%BC%95/%E7%B5%90%E5%90%88%E4%BB%A3%E6%95%B8" rel="nofollow" title="足義友，借鏡照[k]" accesskey="k"><span>援引</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=lzh" title="貢彩件、獻樂謠[u]" accesskey="u"><span>進獻</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%E7%89%B9%E6%AE%8A:%E7%89%B9%E6%9F%A5" title="展盡奇頁[q]" accesskey="q"><span>特查</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E7%B5%90%E5%90%88%E4%BB%A3%E6%95%B8&action=info" title="更多關於此頁面的資訊"><span>文訊</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E6%AE%8A:URL%E7%B8%AE%E7%9F%AD%E7%A8%8B%E5%BC%8F&url=https%3A%2F%2Fzh-classical.wikipedia.org%2Fwiki%2F%25E7%25B5%2590%25E5%2590%2588%25E4%25BB%25A3%25E6%2595%25B8"><span>取得短網址</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a 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href="/wiki/%E8%AA%AA%E6%98%8E:%E6%A0%A1%E5%B0%8D" title="說明:校對">今本</a></b>（此為底本，未經審校）</indicator></div> </div> <div id="siteSub" class="noprint">文出維基大典</div> </div> <div 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>結合代數</b>者，有矢量乘法之<a href="/wiki/%E6%A8%A1_(%E4%BB%A3%E6%95%B8)" title="模 (代數)">模</a>也，然其乘法合<a href="/wiki/%E7%B5%90%E5%90%88%E5%BE%8B" class="mw-redirect" title="結合律">結合律</a>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%B5%90%E5%90%88%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>結合代數</b>者，模也，其標量環為<a href="/wiki/%E7%92%B0_(%E4%BB%A3%E6%95%B8)" title="環 (代數)">交換環</a>，並有<a href="/wiki/%E4%BA%8C%E5%85%83%E9%81%8B%E7%AE%97" title="二元運算">模乘法</a>，合： </p> <ul><li>模合加法，乘法，<a href="/wiki/%E7%92%B0_(%E4%BB%A3%E6%95%B8)" title="環 (代數)">環</a>也。</li> <li>凡標量甲（a）與模之物丙（<b>x</b>）丁（<b>y</b>），有甲丙積乘丁，同乎甲乘丙丁積也（「(a<b>x</b>)(<b>y</b>)=a(<b>xy</b>)。」）</li></ul> <p>若標量環為<a href="/wiki/%E5%9F%9F_(%E4%BB%A3%E6%95%B8)" title="域 (代數)">域</a>，即模為<a href="/wiki/%E7%9F%A2%E9%87%8F%E7%A9%BA%E9%96%93" title="矢量空間">矢量空間</a>者，曰<a href="/wiki/%E4%BB%A3%E6%95%B8_(%E4%BB%A3%E6%95%B8)" title="代數 (代數)">域代數</a>也。 </p><p>若模乘法有<a href="/wiki/%E5%96%AE%E4%BD%8D%E5%85%83" class="mw-redirect" title="單位元">單位元</a>「一」者，曰<b>環代數</b>也。 </p><p><br /> </p> <div class="notice metadata noresize" id="stub" style="font-size: small;"><span typeof="mw:File"><a href="/wiki/%E6%AA%94%E6%A1%88:%E7%B6%AD%E5%9F%BA%E7%9A%84%E3%80%8C%E7%B6%AD%E3%80%8D%E5%AD%97.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/%E7%B6%AD%E5%9F%BA%E7%9A%84%E3%80%8C%E7%B6%AD%E3%80%8D%E5%AD%97.svg/38px-%E7%B6%AD%E5%9F%BA%E7%9A%84%E3%80%8C%E7%B6%AD%E3%80%8D%E5%AD%97.svg.png" decoding="async" width="38" height="38" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/%E7%B6%AD%E5%9F%BA%E7%9A%84%E3%80%8C%E7%B6%AD%E3%80%8D%E5%AD%97.svg/57px-%E7%B6%AD%E5%9F%BA%E7%9A%84%E3%80%8C%E7%B6%AD%E3%80%8D%E5%AD%97.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/%E7%B6%AD%E5%9F%BA%E7%9A%84%E3%80%8C%E7%B6%AD%E3%80%8D%E5%AD%97.svg/76px-%E7%B6%AD%E5%9F%BA%E7%9A%84%E3%80%8C%E7%B6%AD%E3%80%8D%E5%AD%97.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span> <b>結合代數</b>一文似未成。宜<span class="plainlinks"><a class="external text" href="https://zh-classical.wikipedia.org/w/index.php?title=%E7%B5%90%E5%90%88%E4%BB%A3%E6%95%B8&action=edit">善</a></span>之。</div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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