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href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94+%28%D7%9E%D7%91%D7%A0%D7%94+%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת" class=""><span>יצירת חשבון</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=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94+%28%D7%9E%D7%91%D7%A0%D7%94+%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o" class=""><span>כניסה לחשבון</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="אפשרויות נוספות" 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href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&uselang=he"><span>תרומה לוויקיפדיה</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94+%28%D7%9E%D7%91%D7%A0%D7%94+%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>יצירת חשבון</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94+%28%D7%9E%D7%91%D7%A0%D7%94+%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>כניסה לחשבון</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"> דפים לעורכים שלא נכנסו לחשבון <a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D" aria-label="מידע נוסף על עריכה"><span>מידע נוסף</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/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%AA%D7%A8%D7%95%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C%D7%99" title="רשימת העריכות שנעשו מכתובת IP זו [y]" accesskey="y"><span>תרומות</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A9%D7%99%D7%97%D7%94_%D7%A9%D7%9C%D7%99" title="דיון על 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aria-label="תוכן עניינים" 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">תוכן עניינים</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">הסתרה</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">התחלה</div> </a> </li> <li id="toc-הגדרה_פורמלית" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הגדרה_פורמלית"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>הגדרה פורמלית</span> </div> </a> <ul id="toc-הגדרה_פורמלית-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-אסוציאטיביות" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#אסוציאטיביות"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>אסוציאטיביות</span> </div> </a> <ul id="toc-אסוציאטיביות-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קבועי_המבנה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קבועי_המבנה"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>קבועי המבנה</span> </div> </a> <ul id="toc-קבועי_המבנה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-דוגמאות" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#דוגמאות"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>דוגמאות</span> </div> </a> <ul id="toc-דוגמאות-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קבוצת_יוצרים_ואלגברות_אפיניות" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קבוצת_יוצרים_ואלגברות_אפיניות"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>קבוצת יוצרים ואלגברות אפיניות</span> </div> </a> <ul id="toc-קבוצת_יוצרים_ואלגברות_אפיניות-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-איברים_אלגבריים_ושלמים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#איברים_אלגבריים_ושלמים"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>איברים אלגבריים ושלמים</span> </div> </a> <ul id="toc-איברים_אלגבריים_ושלמים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-תורת_המבנה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#תורת_המבנה"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>תורת המבנה</span> </div> </a> <ul id="toc-תורת_המבנה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ראו_גם" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ראו_גם"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>ראו גם</span> </div> </a> <ul id="toc-ראו_גם-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="תוכן עניינים" 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="מצב תוכן העניינים" > <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">מצב תוכן העניינים</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">אלגברה (מבנה אלגברי)</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="מעבר לערך בשפה אחרת. זמין ב־17 שפות" > <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">17 שפות</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Associative_algebra" title="Associative algebra – אנגלית" lang="en" hreflang="en" data-title="Associative algebra" data-language-autonym="English" data-language-local-name="אנגלית" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_associativa" title="Àlgebra associativa – קטלאנית" lang="ca" hreflang="ca" data-title="Àlgebra associativa" data-language-autonym="Català" data-language-local-name="קטלאנית" 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 – גרמנית" lang="de" hreflang="de" data-title="Assoziative Algebra" data-language-autonym="Deutsch" data-language-local-name="גרמנית" class="interlanguage-link-target"><span>Deutsch</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 – אספרנטו" lang="eo" hreflang="eo" data-title="Asocieca alĝebro" data-language-autonym="Esperanto" data-language-local-name="אספרנטו" 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 – ספרדית" lang="es" hreflang="es" data-title="Álgebra asociativa" data-language-autonym="Español" data-language-local-name="ספרדית" 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 – פרסית" lang="fa" hreflang="fa" data-title="جبر شرکت‌پذیر" data-language-autonym="فارسی" data-language-local-name="פרסית" 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 – צרפתית" lang="fr" hreflang="fr" data-title="Algèbre associative" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Asocijativna_algebra" title="Asocijativna algebra – קרואטית" lang="hr" hreflang="hr" data-title="Asocijativna algebra" data-language-autonym="Hrvatski" data-language-local-name="קרואטית" 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 – ‏אינטרלינגואה" lang="ia" hreflang="ia" data-title="Algebra associative" data-language-autonym="Interlingua" data-language-local-name="‏אינטרלינגואה" 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 – אינדונזית" lang="id" hreflang="id" data-title="Aljabar asosiatif" data-language-autonym="Bahasa Indonesia" data-language-local-name="אינדונזית" 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 – יפנית" lang="ja" hreflang="ja" data-title="結合多元環" data-language-autonym="日本語" data-language-local-name="יפנית" 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 – קוריאנית" lang="ko" 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id="mw-content-text" class="mw-body-content"><div class="mw-content-rtl mw-parser-output" lang="he" dir="rtl"><p>ב<a href="/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="מתמטיקה">מתמטיקה</a>, <b>אלגברה מעל חוג</b> היא <a href="/wiki/%D7%9E%D7%95%D7%93%D7%95%D7%9C_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="מודול (מבנה אלגברי)">מודול</a> מעל <a href="/wiki/%D7%97%D7%95%D7%92_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" class="mw-redirect" title="חוג (אלגברה)">חוג</a> <a href="/wiki/%D7%97%D7%99%D7%9C%D7%95%D7%A4%D7%99%D7%95%D7%AA" class="mw-redirect" title="חילופיות">חילופי</a> <a href="/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%91%D7%99%D7%A0%D7%90%D7%A8%D7%99%D7%AA" title="פעולה בינארית">ופעולה בינארית</a> ("כפל") <a href="/wiki/%D7%90%D7%95%D7%A4%D7%A8%D7%98%D7%95%D7%A8_%D7%91%D7%99%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99" title="אופרטור ביליניארי">ביליניארית</a> בין שני איברים שהופכת את המודול לחוג. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="הגדרה_פורמלית"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.94_.D7.A4.D7.95.D7.A8.D7.9E.D7.9C.D7.99.D7.AA"></span>הגדרה פורמלית</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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)&action=edit&section=1" title="עריכת קוד המקור של הפרק: הגדרה פורמלית"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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)&veaction=edit&section=1" title="עריכת פסקה: "הגדרה פורמלית"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>אלגברה היא <a href="/wiki/%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99" title="מבנה אלגברי">מבנה אלגברי</a> הכולל <a href="/wiki/%D7%97%D7%95%D7%92_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" class="mw-redirect" title="חוג (אלגברה)">חוג</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 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> ותת-חוג <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, כך ש-<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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> מוכל ב<a href="/wiki/%D7%9E%D7%A8%D7%9B%D7%96_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%95%D7%92%D7%99%D7%9D)" title="מרכז (תורת החוגים)">מרכז</a> של <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>. הגדרה שקולה: כאשר <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> חוג קומוטטיבי, חוג <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> המהווה מודול מעל <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, ומקיים <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 \alpha (xy)=(\alpha x)y=x(\alpha y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (xy)=(\alpha x)y=x(\alpha y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc49f10932ceb65536417bb52d9b3fe95876c47f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.543ex; height:2.843ex;" alt="{\displaystyle \alpha (xy)=(\alpha x)y=x(\alpha y)}"></span> לכל <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> וסקלר <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 \alpha \in C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eeb6fc73c9737df6db3180a5413bc101298c61a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.095ex; height:2.176ex;" alt="{\displaystyle \alpha \in C}"></span>, נקרא "אלגברה מעל <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>" (לעיתים גם "<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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> - אלגברה"). </p><p>הדוגמה החשובה ביותר היא כאשר <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> הוא <a href="/wiki/%D7%A9%D7%93%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="שדה (מבנה אלגברי)">שדה</a> (ואז המודולים מעל <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> הם מרחבים וקטוריים); <a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">מרחב וקטורי</a> מעל שדה <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> הוא אלגברה, אם מוגדרת בו פעולת כפל ההופכת אותו לחוג (עם יחידה). במקרה כזה ה<a href="/wiki/%D7%9E%D7%9E%D7%93_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="ממד (אלגברה ליניארית)">ממד</a> של האלגברה הוא הממד שלה כמרחב וקטורי. </p><p>כפי שב<a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="אלגברה ליניארית">אלגברה ליניארית</a> נוח לחקור את המרחבים הווקטוריים מעל שדה קבוע ואת הקשרים ביניהם, כך האלגברה המודרנית עוסקת במידה רבה בחקר תכונות של האלגברות השונות מעל חוג בסיס קבוע, ואת הקשרים ביניהן. </p> <div class="mw-heading mw-heading2"><h2 id="אסוציאטיביות"><span id=".D7.90.D7.A1.D7.95.D7.A6.D7.99.D7.90.D7.98.D7.99.D7.91.D7.99.D7.95.D7.AA"></span>אסוציאטיביות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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)&action=edit&section=2" title="עריכת קוד המקור של הפרק: אסוציאטיביות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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)&veaction=edit&section=2" title="עריכת פסקה: "אסוציאטיביות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>בדומה להגדרה של חוג, האלגברה <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> היא '<a href="/wiki/%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%95%D7%AA" class="mw-redirect" title="אסוציאטיביות">אסוציאטיבית</a>' אם <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(yz)=(xy)z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(yz)=(xy)z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b361b83188eb9b2aa2ad507966087aee0b28666d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.864ex; height:2.843ex;" alt="{\displaystyle x(yz)=(xy)z}"></span> לכל <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,z\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fed9181f5a5bbbde61c899c085bdf59577d610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.225ex; height:2.509ex;" alt="{\displaystyle x,y,z\in A}"></span>. לרוב כוללים את דרישת האסוציאטיביות בהגדרה של אלגברה, ואז מגדירים בנפרד מבנה כללי יותר, <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%90_%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%AA" title="אלגברה לא אסוציאטיבית">אלגברה לא אסוציאטיבית</a>, ללא האקסיומה הזו; כך אנו נוהגים בוויקיפדיה. מחברים אחרים אינם דורשים מאלגברה אסוציאטיביות, ואז משתמשים במונח 'אלגברה אסוציאטיבית' עבור אלגברות שכן מקיימות את תכונת האסוציאטיביות. </p><p>בלי קשר להנחת האסוציאטיביות, החוג <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> הוא <b>אלגברה <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> מעל <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span></b> אם <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 \alpha (xy)=(\alpha x)y=x(\alpha y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (xy)=(\alpha x)y=x(\alpha y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc49f10932ceb65536417bb52d9b3fe95876c47f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.543ex; height:2.843ex;" alt="{\displaystyle \alpha (xy)=(\alpha x)y=x(\alpha y)}"></span> לכל <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,\alpha \in C}"> <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>α</mi> <mo>∈</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in A,\alpha \in C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc1cb89d498ac0b3b6afb8fc1516b6e3771c23d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.231ex; height:2.509ex;" alt="{\displaystyle x,y\in A,\alpha \in C}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="קבועי_המבנה"><span id=".D7.A7.D7.91.D7.95.D7.A2.D7.99_.D7.94.D7.9E.D7.91.D7.A0.D7.94"></span>קבועי המבנה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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)&action=edit&section=3" title="עריכת קוד המקור של הפרק: קבועי המבנה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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)&veaction=edit&section=3" title="עריכת פסקה: "קבועי המבנה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>נניח ש-<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> נפרשת כמודול על ידי איברים <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_{1},e_{2},...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},e_{2},...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e73e925d7300a21463985f2bf2924beea969b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.058ex; height:2.009ex;" alt="{\displaystyle e_{1},e_{2},...}"></span>. אז כדי להגדיר את הכפל ב-<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>, מספיק להגדיר את המכפלה <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_{i}\star e_{j}=\sum _{k}\beta _{i,j}^{k}e_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋆</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}\star e_{j}=\sum _{k}\beta _{i,j}^{k}e_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed8bb56ece44e498a334412caac1ca782e7da70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.334ex; height:5.509ex;" alt="{\displaystyle e_{i}\star e_{j}=\sum _{k}\beta _{i,j}^{k}e_{k}}"></span> (כאשר <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 \beta _{i,j}^{k}\in C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>∈</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{i,j}^{k}\in C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9188b0bcc9c577fd3690ac0efd25f70650748b73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.857ex; height:3.509ex;" alt="{\displaystyle \beta _{i,j}^{k}\in C}"></span>). הקבועים <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 \beta _{i,j}^{k}\in C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>∈</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{i,j}^{k}\in C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9188b0bcc9c577fd3690ac0efd25f70650748b73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.857ex; height:3.509ex;" alt="{\displaystyle \beta _{i,j}^{k}\in C}"></span> נקראים <b>קבועי המבנה</b> של <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>. אם <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> הוא <a href="/wiki/%D7%9E%D7%95%D7%93%D7%95%D7%9C_%D7%97%D7%95%D7%A4%D7%A9%D7%99" title="מודול חופשי">מודול חופשי</a> (וזה תמיד כך אם <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> שדה), אז אפשר לתרגם תכונות מסוימות של האלגברה (למשל: אסוציאטיביות) למשוואות על קבועי המבנה; כך אפשר לראות את כל האלגברות האסוציאטיביות מממד <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf8b0f621a23f81aa20d63b5cd59d3dcad83ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle \ n}"></span> מעל שדה נתון כ<a href="/wiki/%D7%99%D7%A8%D7%99%D7%A2%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="יריעה אלגברית">יריעה אלגברית</a> מעל השדה. </p> <div class="mw-heading mw-heading2"><h2 id="דוגמאות"><span id=".D7.93.D7.95.D7.92.D7.9E.D7.90.D7.95.D7.AA"></span>דוגמאות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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)&action=edit&section=4" title="עריכת קוד המקור של הפרק: דוגמאות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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)&veaction=edit&section=4" title="עריכת פסקה: "דוגמאות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>כל חוג הוא אלגברה מעל <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"> <mtext> </mtext> <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/0481c80653b36238e4f4e79d8ab4ed225e5b7fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.131ex; height:2.176ex;" alt="{\displaystyle \ \mathbb {Z} }"></span> או מנה סופית שלו.</li> <li>אם <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/F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K/F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5842a687baef7f99f582b42d6642a8e363b72e15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.969ex; height:2.843ex;" alt="{\displaystyle K/F}"></span> <a href="/wiki/%D7%94%D7%A8%D7%97%D7%91%D7%AA_%D7%A9%D7%93%D7%95%D7%AA" title="הרחבת שדות">הרחבה של שדות</a> (ראו <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%92%D7%9C%D7%95%D7%90%D7%94" title="תורת גלואה">תורת גלואה</a>), אז <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> הוא אלגברה מעל <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>.</li> <li>אוסף <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94" title="מטריצה">המטריצות</a> הריבועיות בגודל <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\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> מעל שדה (או חוג) <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> מהוות אלגברה מעל <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, כשהמכפלה היא <a href="/wiki/%D7%9B%D7%A4%D7%9C_%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%95%D7%AA" title="כפל מטריצות">המכפלה הרגילה של מטריצות</a>.</li> <li><a href="/wiki/%D7%A4%D7%95%D7%9C%D7%99%D7%A0%D7%95%D7%9D#חוג_הפולינומים" title="פולינום">חוג הפולינומים</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 C[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a716ead54fee91528716dc1929b87a6d9aedcc89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.39ex; height:2.843ex;" alt="{\displaystyle C[x]}"></span> הוא אלגברה מעל <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>.</li> <li><a href="/wiki/%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94_%D7%90%D7%95%D7%A7%D7%9C%D7%99%D7%93%D7%99%D7%AA" title="גאומטריה אוקלידית">המרחב האוקלידי</a> התלת-ממדי מהווה אלגברה מעל המספרים הממשיים כאשר <a href="/wiki/%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="מכפלה וקטורית">המכפלה הווקטורית</a> היא המכפלה.</li></ul> <div class="mw-heading mw-heading2"><h2 id="קבוצת_יוצרים_ואלגברות_אפיניות"><span id=".D7.A7.D7.91.D7.95.D7.A6.D7.AA_.D7.99.D7.95.D7.A6.D7.A8.D7.99.D7.9D_.D7.95.D7.90.D7.9C.D7.92.D7.91.D7.A8.D7.95.D7.AA_.D7.90.D7.A4.D7.99.D7.A0.D7.99.D7.95.D7.AA"></span>קבוצת יוצרים ואלגברות אפיניות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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)&action=edit&section=5" title="עריכת קוד המקור של הפרק: קבוצת יוצרים ואלגברות אפיניות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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)&veaction=edit&section=5" title="עריכת פסקה: "קבוצת יוצרים ואלגברות אפיניות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>כל <a href="/wiki/%D7%AA%D7%AA_%D7%A7%D7%91%D7%95%D7%A6%D7%94" class="mw-redirect" title="תת קבוצה">תת קבוצה</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\subset A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c51459534b6075b0c69a18581cc47e34efb0829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.341ex; height:2.176ex;" alt="{\displaystyle S\subset A}"></span> יוצרת את ה<b>תת-אלגברה</b> <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 C[S]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C[S]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/514c2c5b9445a0b7f8448926ae3e24e93d4b7654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.559ex; height:2.843ex;" alt="{\displaystyle C[S]}"></span>, שאבריה מתקבלים מאברי <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> על ידי חיבור, <a href="/wiki/%D7%97%D7%99%D7%A1%D7%95%D7%A8" title="חיסור">חיסור</a>, כפל, וכפל בסקלרים מ-<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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> (זוהי תת-האלגברה הקטנה ביותר של <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> המכילה את <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"> <mtext> </mtext> <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/7bc46c9185d6ff67f4946df6686a8408c9cd1a5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.08ex; height:2.176ex;" alt="{\displaystyle \ S}"></span>). אם קיימת <a href="/wiki/%D7%A7%D7%91%D7%95%D7%A6%D7%94_%D7%A1%D7%95%D7%A4%D7%99%D7%AA" title="קבוצה סופית">קבוצה סופית</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}"> <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> כך ש-<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 C[S]=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C[S]=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c01ca0bedef545d0b365db1d78317cbca37c81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.401ex; height:2.843ex;" alt="{\displaystyle C[S]=A}"></span>, אז <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> <b>נוצרת סופית</b> או <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%90%D7%A4%D7%99%D7%A0%D7%99%D7%AA" title="אלגברה אפינית">אפינית</a>. </p> <div class="mw-heading mw-heading2"><h2 id="איברים_אלגבריים_ושלמים"><span id=".D7.90.D7.99.D7.91.D7.A8.D7.99.D7.9D_.D7.90.D7.9C.D7.92.D7.91.D7.A8.D7.99.D7.99.D7.9D_.D7.95.D7.A9.D7.9C.D7.9E.D7.99.D7.9D"></span>איברים אלגבריים ושלמים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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)&action=edit&section=6" title="עריכת קוד המקור של הפרק: איברים אלגבריים ושלמים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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)&veaction=edit&section=6" title="עריכת פסקה: "איברים אלגבריים ושלמים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>כאשר <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> אלגברה קומוטטיבית, אפשר לחקור את המבנה של האלגברה על ידי סיווג האיברים בהתאם לתכונות שלהם מעל <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. כך איבר <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span> הוא 'אלגברי' אם הוא מקיים <a href="/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94" title="משוואה">משוואה</a> פולינומית עם מקדמים מ-<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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, ו'אלגברי שלם' (או סתם 'שלם') אם הוא מקיים משוואה כזו שהמקדם המוביל שלה הוא <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c043461e59fa0e8d37d8acbd6d7e5b782dd84246" 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 \ 1}"></span>. איברים שאינם אלגבריים נקראים <b>איברים טרנסצנדנטיים</b>; הטרמינולוגיה הזו מכלילה את הדוגמה של המספרים המרוכבים כאלגברה מעל השלמים (ראו <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%98%D7%A8%D7%A0%D7%A1%D7%A6%D7%A0%D7%93%D7%A0%D7%98%D7%99" title="מספר טרנסצנדנטי">מספר טרנסצנדנטי</a>). אם <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> אלגברה מעל <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> שכל איבריה שלמים, אומרים ש-<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/C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87a576f9e68e0e2cb0cd4496d64fa05cf584f3ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.693ex; height:2.843ex;" alt="{\displaystyle R/C}"></span> היא <b>הרחבה שלמה</b>. </p><p>משפט: אלגברה אפינית שהיא שדה מוכרחה להיות אלגברית (ואז היא בעלת ממד סופי). </p><p>משפט הנורמליזציה של <a href="/wiki/%D7%90%D7%9E%D7%99_%D7%A0%D7%AA%D7%A8" title="אמי נתר">נתר</a> קובע שאם <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> אלגברה אפינית מעל שדה <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, אז קיימת תת-אלגברה <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> של <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> שהיא שלמה מעל <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, כך ש-<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> טרנסצנדנטית מעליה (כלומר: האיברים היחידים של <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> שהם אלגבריים מעל <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> הם אברי <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> עצמם). מזה נובע ש<a href="/wiki/%D7%9E%D7%9E%D7%93_%D7%A7%D7%A8%D7%95%D7%9C" title="ממד קרול">ממד קרול</a> של <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> שווה ל<a href="/w/index.php?title=%D7%93%D7%A8%D7%92%D7%AA_%D7%98%D7%A8%D7%A0%D7%A1%D7%A6%D7%A0%D7%93%D7%A0%D7%98%D7%99%D7%95%D7%AA&action=edit&redlink=1" class="new" title="דרגת טרנסצנדנטיות (הדף אינו קיים)">דרגת הטרנסצנדנטיות</a> שלה. </p> <div class="mw-heading mw-heading2"><h2 id="תורת_המבנה"><span id=".D7.AA.D7.95.D7.A8.D7.AA_.D7.94.D7.9E.D7.91.D7.A0.D7.94"></span>תורת המבנה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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)&action=edit&section=7" title="עריכת קוד המקור של הפרק: תורת המבנה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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)&veaction=edit&section=7" title="עריכת פסקה: "תורת המבנה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>המרכז של חוג (אסוציאטיבי) פשוט (עם יחידה) הוא תמיד שדה, ואז אפשר לראות בו אלגברה מעל המרכז. אלגברות פשוטות שהמרכז שלהן הוא שדה קבוע <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> נקראות 'אלגברות פשוטות מרכזיות' מעל <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>. אם הממד סופי, האלגברה היא <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%AA_%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%95%D7%AA" class="mw-redirect" title="אלגברת מטריצות">אלגברת מטריצות</a> מעל <a href="/wiki/%D7%97%D7%95%D7%92_%D7%A2%D7%9D_%D7%97%D7%99%D7%9C%D7%95%D7%A7" title="חוג עם חילוק">אלגברת חילוק</a>. אם <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%A1%D7%92%D7%95%D7%A8_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="שדה סגור אלגברית">סגור אלגברית</a>, אז אין אלגברות חילוק מממד סופי שזה מרכזן, למעט <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> עצמו. </p><p>אם <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> אלגברה <a href="/wiki/%D7%97%D7%95%D7%92_%D7%90%D7%A8%D7%98%D7%99%D7%A0%D7%99" title="חוג ארטיני">ארטינית</a>, אז <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 J(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d0138d6d75330c4628b4dd27b2a0af762a59bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.024ex; height:2.843ex;" alt="{\displaystyle J(A)}"></span> (ה<a href="/wiki/%D7%A8%D7%93%D7%99%D7%A7%D7%9C_%D7%92%27%D7%99%D7%99%D7%A7%D7%95%D7%91%D7%A1%D7%95%D7%9F" title="רדיקל ג'ייקובסון">רדיקל של ג'ייקובסון</a>) הוא <a href="/wiki/%D7%90%D7%99%D7%93%D7%99%D7%90%D7%9C_%D7%A0%D7%99%D7%9C%D7%A4%D7%95%D7%98%D7%A0%D7%98%D7%99" class="mw-redirect" title="אידיאל נילפוטנטי">נילפוטנטי</a>, ו<a href="/wiki/%D7%97%D7%95%D7%92_%D7%9E%D7%A0%D7%94" title="חוג מנה">חוג המנה</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 A/J(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>J</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A/J(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b62be27052744d9c51a2cb96d2286da874d90d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.929ex; height:2.843ex;" alt="{\displaystyle A/J(A)}"></span> הוא אלגברה <a href="/wiki/%D7%97%D7%95%D7%92_%D7%A4%D7%A9%D7%95%D7%98_%D7%9C%D7%9E%D7%97%D7%A6%D7%94" title="חוג פשוט למחצה">פשוטה למחצה</a>, שאפשר לכתוב כ<a href="/wiki/%D7%A1%D7%9B%D7%95%D7%9D_%D7%99%D7%A9%D7%A8" title="סכום ישר">סכום ישר</a> של מספר סופי של אלגברות פשוטות. כל אחת מאלה היא אלגברת מטריצות מעל חוג עם חילוק. </p><p><a href="/w/index.php?title=%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%95%D7%93%D7%A8%D7%91%D7%A8%D7%9F&action=edit&redlink=1" class="new" title="המשפט היסודי של ודרברן (הדף אינו קיים)">המשפט היסודי של ודרברן</a> קובע שאם <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> אלגברה מממד סופי מעל <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%9E%D7%95%D7%A9%D7%9C%D7%9D" title="שדה מושלם">שדה מושלם</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 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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> (ובפרט, מעל <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%A1%D7%92%D7%95%D7%A8_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="שדה סגור אלגברית">שדה סגור אלגברית</a>), אז יש ל-<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> תת-אלגברה <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"> <msup> <mi>A</mi> <mo>′</mo> </msup> </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/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"></span> איזומורפית ל-<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/J(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>J</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A/J(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b62be27052744d9c51a2cb96d2286da874d90d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.929ex; height:2.843ex;" alt="{\displaystyle A/J(A)}"></span>, וקיים פירוק לסכום ישר של מרחבים וקטוריים (אבל לא של חוגים), <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'\oplus J(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>⊕</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=A'\oplus J(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/782d93f4a69aa7417d92ed5eefa0cf4788ba2b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.134ex; height:3.009ex;" alt="{\displaystyle A=A'\oplus J(A)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="ראו_גם"><span id=".D7.A8.D7.90.D7.95_.D7.92.D7.9D"></span>ראו גם</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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)&action=edit&section=8" title="עריכת קוד המקור של הפרק: ראו גם"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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)&veaction=edit&section=8" title="עריכת פסקה: "ראו גם"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%90_%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%AA" title="אלגברה לא אסוציאטיבית">אלגברה לא אסוציאטיבית</a></li></ul> <p><br /> </p> <table class="navbox nowraplinks" style="width: 90%; clear: both; margin: 0.5em auto; margin-top: 0.5em; margin-bottom: 0.5em; padding: 0.2em; text-align: right;"> <tbody><tr> <th colspan="3" style="text-align: center; padding-top: 0.1em; padding-bottom: 0.1em; color: black; background:#d1eeee; font-weight: bold;"><a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9E%D7%95%D7%A4%D7%A9%D7%98%D7%AA" title="אלגברה מופשטת">אלגברה מופשטת</a> </th></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">ענפים </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="אלגברה ליניארית">אלגברה ליניארית</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%91%D7%95%D7%9C%D7%99%D7%90%D7%A0%D7%99%D7%AA" title="אלגברה בוליאנית">אלגברה בוליאנית</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="אלגברה דיפרנציאלית">אלגברה דיפרנציאלית</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%94%D7%95%D7%9E%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%AA" title="אלגברה הומולוגית">אלגברה הומולוגית</a> • <a href="/wiki/%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="גאומטריה אלגברית">גאומטריה אלגברית</a> • <a href="/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="טופולוגיה אלגברית">טופולוגיה אלגברית</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%92%D7%9C%D7%95%D7%90%D7%94" title="תורת גלואה">תורת גלואה</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA" title="תורת החבורות">תורת החבורות</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%95%D7%92%D7%99%D7%9D" title="תורת החוגים">תורת החוגים</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="תורת המספרים האלגברית">תורת המספרים האלגברית</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%95%D7%AA" title="תורת הקטגוריות">תורת הקטגוריות</a> • <a href="/w/index.php?title=%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A9%D7%93%D7%95%D7%AA&action=edit&redlink=1" class="new" title="תורת השדות (הדף אינו קיים)">תורת השדות</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;"><a href="/wiki/%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99" title="מבנה אלגברי">מבנים אלגבריים</a> </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9E%D7%90%D7%92%D7%9E%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="מאגמה (מבנה אלגברי)">מאגמה</a> • <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%9C%D7%9E%D7%97%D7%A6%D7%94" title="חבורה למחצה">חבורה למחצה</a> • <a href="/wiki/%D7%9E%D7%95%D7%A0%D7%95%D7%90%D7%99%D7%93_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="מונואיד (מבנה אלגברי)">מונואיד</a> • <a href="/wiki/%D7%97%D7%91%D7%95%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="חבורה (מבנה אלגברי)">חבורה</a> • <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%90%D7%91%D7%9C%D7%99%D7%AA" title="חבורה אבלית">חבורה אַבּלִית</a> • <a href="/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)">חוג</a> • <a href="/wiki/%D7%AA%D7%97%D7%95%D7%9D_%D7%A9%D7%9C%D7%9E%D7%95%D7%AA" title="תחום שלמות">תחום שלמות</a> • <a href="/wiki/%D7%A9%D7%93%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="שדה (מבנה אלגברי)">שדה</a> • <a href="/wiki/%D7%9E%D7%95%D7%93%D7%95%D7%9C_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="מודול (מבנה אלגברי)">מודול</a> • <a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">מרחב וקטורי</a> • <a class="mw-selflink selflink">אלגברה (מבנה אלגברי)</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%AA_%D7%9C%D7%99" title="אלגברת לי">אלגברת לי</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%AA_%D7%94%D7%A7%D7%95%D7%95%D7%98%D7%A8%D7%A0%D7%99%D7%95%D7%A0%D7%99%D7%9D_%D7%A9%D7%9C_%D7%94%D7%9E%D7%99%D7%9C%D7%98%D7%95%D7%9F" title="אלגברת הקווטרניונים של המילטון">אלגברת הקווטרניונים של המילטון</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%90_%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%AA" title="אלגברה לא אסוציאטיבית">אלגברה לא אסוציאטיבית</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">מושגי יסוד </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%94%D7%95%D7%9E%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="הומומורפיזם">הומומורפיזם</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98%D7%99_%D7%94%D7%90%D7%99%D7%96%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="משפטי האיזומורפיזם">משפטי האיזומורפיזם</a> • <a href="/wiki/%D7%AA%D7%AA-%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99%D7%AA" title="תת-חבורה נורמלית">תת-חבורה נורמלית</a> • <a href="/wiki/%D7%90%D7%99%D7%93%D7%99%D7%90%D7%9C_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="אידיאל (אלגברה)">אידיאל</a> • <a href="/wiki/%D7%9C%D7%95%D7%A7%D7%9C%D7%99%D7%96%D7%A6%D7%99%D7%94_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%95%D7%92%D7%99%D7%9D)" title="לוקליזציה (תורת החוגים)">לוקליזציה</a> • <a href="/wiki/%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%98%D7%A0%D7%96%D7%95%D7%A8%D7%99%D7%AA" title="מכפלה טנזורית">מכפלה טנזורית</a> • <a href="/wiki/%D7%94%D7%A6%D7%92%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="הצגה ליניארית">הצגה ליניארית</a> </td></tr> </tbody></table> <div role="navigation" class="navbox authority-control" aria-labelledby="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q744960?uselang=he&#124;עריכת_הנתון_בוויקינתונים" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q744960?uselang=he&#124;עריכת_הנתון_בוויקינתונים" scope="row" class="navbox-group" style="width:1%"><a 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href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA" title="הספרייה הלאומית">NLI</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007295882805171">987007295882805171</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5_%D7%91%D7%A7%D7%A8%D7%94_%D7%9E%D7%A9%D7%95%D7%9C%D7%91" title="קובץ בקרה משולב">GND</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4293934-3">4293934-3</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%A9%D7%99%D7%98%D7%AA_%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%AA_%D7%94%D7%A7%D7%95%D7%A0%D7%92%D7%A8%D7%A1" title="שיטת ספריית הקונגרס">LCCN</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85008821">sh85008821</a></span></span></li></ul> </div></div></td></tr></tbody></table></div>    </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="">אוחזר מתוך "<a dir="ltr" href="https://he.wikipedia.org/w/index.php?title=אלגברה_(מבנה_אלגברי)&oldid=38783270">https://he.wikipedia.org/w/index.php?title=אלגברה_(מבנה_אלגברי)&oldid=38783270</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94" title="ויקיפדיה:קטגוריה">קטגוריה</a>: <ul><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%9E%D7%91%D7%A0%D7%99%D7%9D_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%99%D7%9D" title="קטגוריה:מבנים אלגבריים">מבנים אלגבריים</a></li></ul></div><div id="mw-hidden-catlinks" 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