CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Module (mathematics) - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"2d0c0a2c-e03b-436b-910c-4a215661b50f","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Module_(mathematics)","wgTitle":"Module (mathematics)","wgCurRevisionId":1251812964,"wgRevisionId":1251812964,"wgArticleId":276410,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Articles lacking in-text citations from May 2015","All articles lacking in-text citations","All articles with unsourced statements","Articles with unsourced statements from May 2015","Algebraic structures","Module theory"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Module_(mathematics)","wgRelevantArticleId":276410, "wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q18848","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true, "wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap", "ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Module (mathematics) - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Module_(mathematics)"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Module_(mathematics)&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Module_(mathematics)"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Module_mathematics rootpage-Module_mathematics skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Module+%28mathematics%29" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</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=Special:UserLogin&returnto=Module+%28mathematics%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</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="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Module+%28mathematics%29" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Module+%28mathematics%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</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"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</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/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" 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">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</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">(Top)</div> </a> </li> <li id="toc-Introduction_and_definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Introduction_and_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Introduction and definition</span> </div> </a> <button aria-controls="toc-Introduction_and_definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Introduction and definition subsection</span> </button> <ul id="toc-Introduction_and_definition-sublist" class="vector-toc-list"> <li id="toc-Motivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Motivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Motivation</span> </div> </a> <ul id="toc-Motivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Formal definition</span> </div> </a> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Submodules_and_homomorphisms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Submodules_and_homomorphisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Submodules and homomorphisms</span> </div> </a> <ul id="toc-Submodules_and_homomorphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types_of_modules" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_modules"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Types of modules</span> </div> </a> <ul id="toc-Types_of_modules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_notions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_notions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Further notions</span> </div> </a> <button aria-controls="toc-Further_notions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Further notions subsection</span> </button> <ul id="toc-Further_notions-sublist" class="vector-toc-list"> <li id="toc-Relation_to_representation_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_representation_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Relation to representation theory</span> </div> </a> <ul id="toc-Relation_to_representation_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Module (mathematics)</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="Go to an article in another language. Available in 39 languages" > <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-39" 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">39 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D8%AD%D9%84%D9%82%D9%8A" title="فضاء حلقي – Arabic" lang="ar" hreflang="ar" data-title="فضاء حلقي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D2%A0%D1%83%D0%BB%D1%81%D0%B0_%D3%A9%D2%AB%D1%82%D3%A9%D0%BD%D0%B4%D3%99_%D0%BC%D0%BE%D0%B4%D1%83%D0%BB%D1%8C" title="Ҡулса өҫтөндә модуль – Bashkir" lang="ba" hreflang="ba" data-title="Ҡулса өҫтөндә модуль" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D1%83%D0%BB_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BF%D1%80%D1%8A%D1%81%D1%82%D0%B5%D0%BD%D0%B8%D1%82%D0%B5)" title="Модул (теория на пръстените) – Bulgarian" lang="bg" hreflang="bg" data-title="Модул (теория на пръстените)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/M%C3%B2dul" title="Mòdul – Catalan" lang="ca" hreflang="ca" data-title="Mòdul" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Modul_(matematika)" title="Modul (matematika) – Czech" lang="cs" hreflang="cs" data-title="Modul (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Modul_(Mathematik)" title="Modul (Mathematik) – German" lang="de" hreflang="de" data-title="Modul (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Moodul_(algebra)" title="Moodul (algebra) – Estonian" lang="et" hreflang="et" data-title="Moodul (algebra)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CF%8C%CF%84%CF%85%CF%80%CE%BF_(%CE%AC%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1)" title="Πρότυπο (άλγεβρα) – Greek" lang="el" hreflang="el" data-title="Πρότυπο (άλγεβρα)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/M%C3%B3dulo_(matem%C3%A1tica)" title="Módulo (matemática) – Spanish" lang="es" hreflang="es" data-title="Módulo (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Modulo_(algebro)" title="Modulo (algebro) – Esperanto" lang="eo" hreflang="eo" data-title="Modulo (algebro)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Modulu_(matematika)" title="Modulu (matematika) – Basque" lang="eu" hreflang="eu" data-title="Modulu (matematika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AF%D9%88%D9%84_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="مدول (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="مدول (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Persian" 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/Module_sur_un_anneau" title="Module sur un anneau – French" lang="fr" hreflang="fr" data-title="Module sur un anneau" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/M%C3%B3dulo_(%C3%A1lxebra)" title="Módulo (álxebra) – Galician" lang="gl" hreflang="gl" data-title="Módulo (álxebra)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%80%EA%B5%B0" title="가군 – Korean" lang="ko" hreflang="ko" data-title="가군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%B8%D5%A4%D5%B8%D6%82%D5%AC_%D6%85%D5%B2%D5%A1%D5%AF%D5%AB_%D5%BE%D6%80%D5%A1" title="Մոդուլ օղակի վրա – Armenian" lang="hy" hreflang="hy" data-title="Մոդուլ օղակի վրա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Modul_(algebra)" title="Modul (algebra) – Croatian" lang="hr" hreflang="hr" data-title="Modul (algebra)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Modul_(matematika)" title="Modul (matematika) – Indonesian" lang="id" hreflang="id" data-title="Modul (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Modulo_(algebra)" title="Modulo (algebra) – Interlingua" lang="ia" hreflang="ia" data-title="Modulo (algebra)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Modulo_(algebra)" title="Modulo (algebra) – Italian" lang="it" hreflang="it" data-title="Modulo (algebra)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/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) – Hebrew" lang="he" hreflang="he" data-title="מודול (מבנה אלגברי)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Modulus_(matematika)" title="Modulus (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Modulus (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Moduul" title="Moduul – Dutch" lang="nl" hreflang="nl" data-title="Moduul" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%92%B0%E4%B8%8A%E3%81%AE%E5%8A%A0%E7%BE%A4" title="環上の加群 – Japanese" lang="ja" hreflang="ja" data-title="環上の加群" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Modul_(matematikk)" title="Modul (matematikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Modul (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Modul_i_matematikk" title="Modul i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Modul i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AE%E0%A9%8C%E0%A8%A1%E0%A8%BF%E0%A8%8A%E0%A8%B2" title="ਮੌਡਿਊਲ – Punjabi" lang="pa" hreflang="pa" data-title="ਮੌਡਿਊਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Modu%C5%82_(matematyka)" title="Moduł (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Moduł (matematyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/M%C3%B3dulo_(%C3%A1lgebra)" title="Módulo (álgebra) – Portuguese" lang="pt" hreflang="pt" data-title="Módulo (álgebra)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D1%83%D0%BB%D1%8C_%D0%BD%D0%B0%D0%B4_%D0%BA%D0%BE%D0%BB%D1%8C%D1%86%D0%BE%D0%BC" title="Модуль над кольцом – Russian" lang="ru" hreflang="ru" data-title="Модуль над кольцом" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Module_(mathematics)" title="Module (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Module (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr badge-Q70893996 mw-list-item" title=""><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D1%83%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Модул (математика) – Serbian" lang="sr" hreflang="sr" data-title="Модул (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Moduli_(algebra)" title="Moduli (algebra) – Finnish" lang="fi" hreflang="fi" data-title="Moduli (algebra)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Modul_(matematik)" title="Modul (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Modul (matematik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mod%C3%BCl_(matematik)" title="Modül (matematik) – Turkish" lang="tr" hreflang="tr" data-title="Modül (matematik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D1%83%D0%BB%D1%8C_%D0%BD%D0%B0%D0%B4_%D0%BA%D1%96%D0%BB%D1%8C%D1%86%D0%B5%D0%BC" title="Модуль над кільцем – Ukrainian" lang="uk" hreflang="uk" data-title="Модуль над кільцем" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%A8%A1_(%E4%BB%A3%E6%95%B8)" title="模 (代數) – Literary Chinese" lang="lzh" hreflang="lzh" data-title="模 (代數)" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%A8%A1_(%E4%BB%A3%E6%95%B8)" title="模 (代數) – Cantonese" lang="yue" hreflang="yue" data-title="模 (代數)" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A8%A1" title="模 – Chinese" lang="zh" hreflang="zh" data-title="模" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q18848#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Module_(mathematics)" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Module_(mathematics)" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Module_(mathematics)"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Module_(mathematics)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Module_(mathematics)&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Module_(mathematics)"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Module_(mathematics)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Module_(mathematics)&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Module_(mathematics)" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Module_(mathematics)" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Module_(mathematics)&oldid=1251812964" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Module_(mathematics)&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Module_%28mathematics%29&id=1251812964&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FModule_%28mathematics%29"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FModule_%28mathematics%29"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Module_%28mathematics%29&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Module_(mathematics)&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/Abstract_Algebra/Modules" hreflang="en"><span>Wikibooks</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q18848" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Generalization of vector spaces from fields to rings</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">May 2015</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist" style="width: 20.5em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory</span><br /><a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Basic concepts</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><b><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Rings</a></b> <dl><dd>• <a href="/wiki/Subring" title="Subring">Subrings</a></dd> <dd>• <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">Ideal</a></dd> <dd>• <a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a> <dl><dd>• <a href="/wiki/Fractional_ideal" title="Fractional ideal">Fractional ideal</a></dd> <dd>• <a href="/wiki/Total_ring_of_fractions" title="Total ring of fractions">Total ring of fractions</a></dd></dl></dd> <dd>• <a href="/wiki/Product_of_rings" title="Product of rings">Product of rings</a></dd> <dd>• <a href="/wiki/Free_product_of_associative_algebras" title="Free product of associative algebras">Free product of associative algebras</a></dd> <dd>• <a href="/wiki/Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a></dd></dl> <p><b><a href="/wiki/Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a></b> </p> <dl><dd>• <a href="/wiki/Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">Kernel</a></dd> <dd>• <a href="/wiki/Inner_automorphism#Ring_case" title="Inner automorphism">Inner automorphism</a></dd> <dd>• <a href="/wiki/Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></dd></dl> <p><b><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a></b> </p> <dl><dd>• <a class="mw-selflink selflink">Module</a></dd> <dd>• <a href="/wiki/Associative_algebra" title="Associative algebra">Associative algebra</a></dd> <dd>• <a href="/wiki/Graded_ring" title="Graded ring">Graded ring</a></dd> <dd>• <a href="/wiki/Involutive_ring" class="mw-redirect" title="Involutive ring">Involutive ring</a></dd> <dd>• <a href="/wiki/Category_of_rings" title="Category of rings">Category of rings</a> <dl><dd>• <a href="/wiki/Integer" title="Integer">Initial ring</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span></dd> <dd>• <a href="/wiki/Zero_ring" title="Zero ring">Terminal ring</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 0=\mathbb {Z} /1\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45ab495cb8cfbac68a9322af662c3d6c7dbe494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:2.843ex;" alt="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"></span></dd></dl></dd></dl> <p><b>Related structures</b> </p> <dl><dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Non-associative_ring" class="mw-redirect" title="Non-associative ring">Non-associative ring</a> <dl><dd>• <a href="/wiki/Lie_ring" class="mw-redirect" title="Lie ring">Lie ring</a></dd> <dd>• <a href="/wiki/Jordan_ring" class="mw-redirect" title="Jordan ring">Jordan ring</a></dd></dl></dd> <dd>• <a href="/wiki/Semiring" title="Semiring">Semiring</a> <dl><dd>• <a href="/wiki/Semifield" title="Semifield">Semifield</a></dd></dl></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><b><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative rings</a></b> <dl><dd>• <a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a> <dl><dd>• <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">Integrally closed domain</a></dd> <dd>• <a href="/wiki/GCD_domain" title="GCD domain">GCD domain</a></dd> <dd>• <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a></dd> <dd>• <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a></dd> <dd>• <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a></dd> <dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a></dd> <dd>• <a href="/wiki/Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">Formal power series ring</a></dd></dl></dd></dl> <p><b><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></b> </p> <dl><dd>• <a href="/wiki/Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></dd> <dd>• <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo <span class="texhtml mvar" style="font-style:italic;">n</span></a></dd> <dd>• <a href="/wiki/Ring_of_integers" title="Ring of integers">Ring of integers</a></dd> <dd>• <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer"><i>p</i>-adic integers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"></span></dd> <dd>• <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"></span></dd> <dd>• <a href="/wiki/Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer <i>p</i>-ring</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} (p^{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} (p^{\infty })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14af623e08c241266c125ad927dd35086ec8ce90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.404ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} (p^{\infty })}"></span></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><b><a href="/wiki/Noncommutative_ring" title="Noncommutative ring">Noncommutative rings</a></b> <dl><dd>• <a href="/wiki/Division_ring" title="Division ring">Division ring</a></dd> <dd>• <a href="/wiki/Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></dd> <dd>• <a href="/wiki/Simple_ring" title="Simple ring">Simple ring</a></dd> <dd>• <a href="/wiki/Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator</a></dd></dl> <p><b><a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></b> </p><p><b><a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a></b> </p><p><b><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></b> </p> <dl><dd>• <a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl> <b><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></b></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Ring_theory_sidebar" title="Template:Ring theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Ring_theory_sidebar" title="Template talk:Ring theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Ring_theory_sidebar" title="Special:EditPage/Template:Ring theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="display:block;margin-bottom:0.35em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a></li> <li><a href="/wiki/Semigroup" title="Semigroup">Semigroup</a> / <a href="/wiki/Monoid" title="Monoid">Monoid</a></li> <li><a href="/wiki/Racks_and_quandles" title="Racks and quandles">Rack and quandle</a></li> <li><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup and loop</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a></li> <li><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li></ul> </div> <i><a href="/wiki/Group_theory" title="Group theory">Group theory</a></i></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a></li> <li><a href="/wiki/Rng_(algebra)" title="Rng (algebra)">Rng</a></li> <li><a href="/wiki/Semiring" title="Semiring">Semiring</a></li> <li><a href="/wiki/Near-ring" title="Near-ring">Near-ring</a></li> <li><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative ring</a></li> <li><a href="/wiki/Domain_(ring_theory)" title="Domain (ring theory)">Domain</a></li> <li><a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a></li> <li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a></li> <li><a href="/wiki/Division_ring" title="Division ring">Division ring</a></li> <li><a href="/wiki/Lie_algebra#Lie_ring" title="Lie algebra">Lie ring</a></li></ul> </div> <i><a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></i></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented lattice</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li></ul> </div> <ul><li><a href="/wiki/Map_of_lattices" title="Map of lattices">Map of lattices</a></li> <li><i><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice theory</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Module</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a class="mw-selflink selflink">Module</a></li> <li><a href="/wiki/Group_with_operators" title="Group with operators">Group with operators</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div> <ul><li><i><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Associative_algebra" title="Associative algebra">Associative</a></li> <li><a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative</a></li> <li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a></li> <li><a href="/wiki/Graded_ring" title="Graded ring">Graded</a></li> <li><a href="/wiki/Bialgebra" title="Bialgebra">Bialgebra</a></li> <li><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Algebraic_structures" title="Template:Algebraic structures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_structures" title="Template talk:Algebraic structures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_structures" title="Special:EditPage/Template:Algebraic structures"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>module</b> is a generalization of the notion of <a href="/wiki/Vector_space" title="Vector space">vector space</a> in which the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a> is replaced by a (not necessarily <a href="/wiki/Commutative_ring" title="Commutative ring">commutative</a>) <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>. The concept of a <i>module</i> also generalizes the notion of an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>, since the abelian groups are exactly the modules over the ring of <a href="/wiki/Integer" title="Integer">integers</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Like a vector space, a module is an additive abelian group, and scalar multiplication is <a href="/wiki/Distributive_property" title="Distributive property">distributive</a> over the operations of addition between elements of the ring or module and is <a href="/wiki/Semigroup_action" title="Semigroup action">compatible</a> with the ring multiplication. </p><p>Modules are very closely related to the <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a> of <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>. They are also one of the central notions of <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a> and <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>, and are used widely in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> and <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction_and_definition">Introduction and definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=1" title="Edit section: Introduction and definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Motivation">Motivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=2" title="Edit section: Motivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a vector space, the set of <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a> is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> and acts on the vectors by scalar multiplication, subject to certain axioms such as the <a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributive law</a>. In a module, the scalars need only be a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, so the module concept represents a significant generalization. In commutative algebra, both <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a> and <a href="/wiki/Quotient_ring" title="Quotient ring">quotient rings</a> are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules. </p><p>Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "<a href="/wiki/Well-behaved" class="mw-redirect" title="Well-behaved">well-behaved</a>" ring, such as a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>, and even for those that do (<a href="/wiki/Free_module" title="Free module">free modules</a>) the number of elements in a basis need not be the same for all bases (that is to say that they may not have a unique <a href="/wiki/Free_module#Definition" title="Free module">rank</a>) if the underlying ring does not satisfy the <a href="/wiki/Invariant_basis_number" title="Invariant basis number">invariant basis number</a> condition, unlike vector spaces, which always have a (possibly infinite) basis whose <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> is then unique. (These last two assertions require the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> in general, but not in the case of <a href="/wiki/Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a> vector spaces, or certain well-behaved infinite-dimensional vector spaces such as <a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> spaces</a>.) </p> <div class="mw-heading mw-heading3"><h3 id="Formal_definition">Formal definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=3" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that <i>R</i> is a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, and 1 is its multiplicative identity. A <b>left <i>R</i>-module</b> <i>M</i> consists of an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> <span class="nowrap">(<i>M</i>, +)</span> and an operation <span class="nowrap"><b>·</b> : <i>R</i> × <i>M</i> → <i>M</i></span> such that for all <i>r</i>, <i>s</i> in <i>R</i> and <i>x</i>, <i>y</i> in <i>M</i>, we have </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\cdot (x+y)=r\cdot x+r\cdot y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <mi>r</mi> <mo>⋅</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\cdot (x+y)=r\cdot x+r\cdot y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e135713749ad83932bd05eae55a4788a5e3412c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.742ex; height:2.843ex;" alt="{\displaystyle r\cdot (x+y)=r\cdot x+r\cdot y}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r+s)\cdot x=r\cdot x+s\cdot x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <mo>⋅</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r+s)\cdot x=r\cdot x+s\cdot x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f8981e6a9c302ab83827b0438bc17f6e86796c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.893ex; height:2.843ex;" alt="{\displaystyle (r+s)\cdot x=r\cdot x+s\cdot x}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (rs)\cdot x=r\cdot (s\cdot x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mi>s</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>⋅</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (rs)\cdot x=r\cdot (s\cdot x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4d4274dd9abe9c72744668b47f8c6e2d86c05d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.692ex; height:2.843ex;" alt="{\displaystyle (rs)\cdot x=r\cdot (s\cdot x)}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot x=x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot x=x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df6fbe6791389457f3711e555c8f46ce20defc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.246ex; height:2.176ex;" alt="{\displaystyle 1\cdot x=x.}"></span></li></ol> <p>The operation · is called <i>scalar multiplication</i>. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in <i>R</i>. One may write <sub><i>R</i></sub><i>M</i> to emphasize that <i>M</i> is a left <i>R</i>-module. A <b>right <i>R</i>-module</b> <i>M</i><sub><i>R</i></sub> is defined similarly in terms of an operation <span class="nowrap">· : <i>M</i> × <i>R</i> → <i>M</i></span>. </p><p>Authors who do not require rings to be <a href="/wiki/Unital_algebra" class="mw-redirect" title="Unital algebra">unital</a> omit condition 4 in the definition above; they would call the structures defined above "unital left <i>R</i>-modules". In this article, consistent with the <a href="/wiki/Glossary_of_ring_theory" title="Glossary of ring theory">glossary of ring theory</a>, all rings and modules are assumed to be unital.<sup id="cite_ref-DummitFoote_2-0" class="reference"><a href="#cite_note-DummitFoote-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>An (<i>R</i>,<i>S</i>)-<a href="/wiki/Bimodule" title="Bimodule">bimodule</a> is an abelian group together with both a left scalar multiplication · by elements of <i>R</i> and a right scalar multiplication ∗ by elements of <i>S</i>, making it simultaneously a left <i>R</i>-module and a right <i>S</i>-module, satisfying the additional condition <span class="nowrap">(<i>r</i> · <i>x</i>) ∗ <i>s</i> = <i>r</i> ⋅ (<i>x</i> ∗ <i>s</i>)</span> for all <i>r</i> in <i>R</i>, <i>x</i> in <i>M</i>, and <i>s</i> in <i>S</i>. </p><p>If <i>R</i> is <a href="/wiki/Commutative_ring" title="Commutative ring">commutative</a>, then left <i>R</i>-modules are the same as right <i>R</i>-modules and are simply called <i>R</i>-modules. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=4" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>If <i>K</i> is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, then <i>K</i>-modules are called <i>K</i>-<a href="/wiki/Vector_space" title="Vector space">vector spaces</a> (vector spaces over <i>K</i>).</li> <li>If <i>K</i> is a field, and <i>K</i>[<i>x</i>] a univariate <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a>, then a <a href="/wiki/Polynomial_ring#Modules" title="Polynomial ring"><i>K</i>[<i>x</i>]-module</a> <i>M</i> is a <i>K</i>-module with an additional action of <i>x</i> on <i>M</i> by a group homomorphism that commutes with the action of <i>K</i> on <i>M</i>. In other words, a <i>K</i>[<i>x</i>]-module is a <i>K</i>-vector space <i>M</i> combined with a <a href="/wiki/Linear_map" title="Linear map">linear map</a> from <i>M</i> to <i>M</i>. Applying the <a href="/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain" title="Structure theorem for finitely generated modules over a principal ideal domain">structure theorem for finitely generated modules over a principal ideal domain</a> to this example shows the existence of the <a href="/wiki/Rational_canonical_form" class="mw-redirect" title="Rational canonical form">rational</a> and <a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan canonical</a> forms.</li> <li>The concept of a <b>Z</b>-module agrees with the notion of an abelian group. That is, every <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> is a module over the ring of <a href="/wiki/Integer" title="Integer">integers</a> <b>Z</b> in a unique way. For <span class="nowrap"><i>n</i> > 0</span>, let <span class="nowrap"><i>n</i> ⋅ <i>x</i> = <i>x</i> + <i>x</i> + ... + <i>x</i></span> (<i>n</i> summands), <span class="nowrap">0 ⋅ <i>x</i> = 0</span>, and <span class="nowrap">(−<i>n</i>) ⋅ <i>x</i> = −(<i>n</i> ⋅ <i>x</i>)</span>. Such a module need not have a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>—groups containing <a href="/wiki/Torsion_element" class="mw-redirect" title="Torsion element">torsion elements</a> do not. (For example, in the group of integers <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> 3, one cannot find even one element that satisfies the definition of a <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> set, since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a <a href="/wiki/Finite_field" title="Finite field">finite field</a> is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)</li> <li>The <a href="/wiki/Decimal_fractions" class="mw-redirect" title="Decimal fractions">decimal fractions</a> (including negative ones) form a module over the integers. Only <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">singletons</a> are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank.</li> <li>If <i>R</i> is any ring and <i>n</i> a <a href="/wiki/Natural_number" title="Natural number">natural number</a>, then the <a href="/wiki/Cartesian_product" title="Cartesian product">cartesian product</a> <i>R</i><sup><i>n</i></sup> is both a left and right <i>R</i>-module over <i>R</i> if we use the component-wise operations. Hence when <span class="nowrap"><i>n</i> = 1</span>, <i>R</i> is an <i>R</i>-module, where the scalar multiplication is just ring multiplication. The case <span class="nowrap"><i>n</i> = 0</span> yields the trivial <i>R</i>-module {0} consisting only of its identity element. Modules of this type are called <a href="/wiki/Free_module" title="Free module">free</a> and if <i>R</i> has <a href="/wiki/Invariant_basis_number" title="Invariant basis number">invariant basis number</a> (e.g. any commutative ring or field) the number <i>n</i> is then the rank of the free module.</li> <li>If M<sub><i>n</i></sub>(<i>R</i>) is the ring of <span class="nowrap"><i>n</i> × <i>n</i></span> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> over a ring <i>R</i>, <i>M</i> is an M<sub><i>n</i></sub>(<i>R</i>)-module, and <i>e</i><sub><i>i</i></sub> is the <span class="nowrap"><i>n</i> × <i>n</i></span> matrix with 1 in the <span class="nowrap">(<i>i</i>, <i>i</i>)</span>-entry (and zeros elsewhere), then <i>e</i><sub><i>i</i></sub><i>M</i> is an <i>R</i>-module, since <span class="nowrap"><i>re</i><sub><i>i</i></sub><i>m</i> = <i>e</i><sub><i>i</i></sub><i>rm</i> ∈ <i>e</i><sub><i>i</i></sub><i>M</i></span>. So <i>M</i> breaks up as the <a href="/wiki/Direct_sum" title="Direct sum">direct sum</a> of <i>R</i>-modules, <span class="nowrap"><i>M</i> = <i>e</i><sub>1</sub><i>M</i> ⊕ ... ⊕ <i>e</i><sub><i>n</i></sub><i>M</i></span>. Conversely, given an <i>R</i>-module <i>M</i><sub>0</sub>, then <i>M</i><sub>0</sub><sup>⊕<i>n</i></sup> is an M<sub><i>n</i></sub>(<i>R</i>)-module. In fact, the <a href="/wiki/Category_of_modules" title="Category of modules">category of <i>R</i>-modules</a> and the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of M<sub><i>n</i></sub>(<i>R</i>)-modules are <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalent</a>. The special case is that the module <i>M</i> is just <i>R</i> as a module over itself, then <i>R</i><sup><i>n</i></sup> is an M<sub><i>n</i></sub>(<i>R</i>)-module.</li> <li>If <i>S</i> is a <a href="/wiki/Empty_set" title="Empty set">nonempty</a> <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, <i>M</i> is a left <i>R</i>-module, and <i>M</i><sup><i>S</i></sup> is the collection of all <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> <span class="nowrap"><i>f</i> : <i>S</i> → <i>M</i></span>, then with addition and scalar multiplication in <i>M</i><sup><i>S</i></sup> defined pointwise by <span class="nowrap">(<i>f</i> + <i>g</i>)(<i>s</i>) = <i>f</i>(<i>s</i>) + <i>g</i>(<i>s</i>)</span> and <span class="nowrap">(<i>rf</i>)(<i>s</i>) = <i>rf</i>(<i>s</i>)</span>, <i>M</i><sup><i>S</i></sup> is a left <i>R</i>-module. The right <i>R</i>-module case is analogous. In particular, if <i>R</i> is commutative then the collection of <i>R-module homomorphisms</i> <span class="nowrap"><i>h</i> : <i>M</i> → <i>N</i></span> (see below) is an <i>R</i>-module (and in fact a <i>submodule</i> of <i>N</i><sup><i>M</i></sup>).</li> <li>If <i>X</i> is a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>, then the <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> from <i>X</i> to the <a href="/wiki/Real_number" title="Real number">real numbers</a> form a ring <i>C</i><sup>∞</sup>(<i>X</i>). The set of all smooth <a href="/wiki/Vector_field" title="Vector field">vector fields</a> defined on <i>X</i> forms a module over <i>C</i><sup>∞</sup>(<i>X</i>), and so do the <a href="/wiki/Tensor_field" title="Tensor field">tensor fields</a> and the <a href="/wiki/Differential_form" title="Differential form">differential forms</a> on <i>X</i>. More generally, the sections of any <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> form a <a href="/wiki/Projective_module" title="Projective module">projective module</a> over <i>C</i><sup>∞</sup>(<i>X</i>), and by <a href="/wiki/Swan%27s_theorem" class="mw-redirect" title="Swan's theorem">Swan's theorem</a>, every projective module is isomorphic to the module of sections of some vector bundle; the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of <i>C</i><sup>∞</sup>(<i>X</i>)-modules and the category of vector bundles over <i>X</i> are <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalent</a>.</li> <li>If <i>R</i> is any ring and <i>I</i> is any <a href="/wiki/Ring_ideal" class="mw-redirect" title="Ring ideal">left ideal</a> in <i>R</i>, then <i>I</i> is a left <i>R</i>-module, and analogously right ideals in <i>R</i> are right <i>R</i>-modules.</li> <li>If <i>R</i> is a ring, we can define the <a href="/wiki/Opposite_ring" title="Opposite ring">opposite ring</a> <i>R</i><sup>op</sup>, which has the same <a href="/wiki/Underlying_set" class="mw-redirect" title="Underlying set">underlying set</a> and the same addition operation, but the opposite multiplication: if <span class="nowrap"><i>ab</i> = <i>c</i></span> in <i>R</i>, then <span class="nowrap"><i>ba</i> = <i>c</i></span> in <i>R</i><sup>op</sup>. Any <i>left</i> <i>R</i>-module <i>M</i> can then be seen to be a <i>right</i> module over <i>R</i><sup>op</sup>, and any right module over <i>R</i> can be considered a left module over <i>R</i><sup>op</sup>.</li> <li><a href="/wiki/Glossary_of_Lie_algebras#Representation_theory" class="mw-redirect" title="Glossary of Lie algebras">Modules over a Lie algebra</a> are (associative algebra) modules over its <a href="/wiki/Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a>.</li> <li>If <i>R</i> and <i>S</i> are rings with a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> <span class="nowrap"><i>φ</i> : <i>R</i> → <i>S</i></span>, then every <i>S</i>-module <i>M</i> is an <i>R</i>-module by defining <span class="nowrap"><i>rm</i> = <i>φ</i>(<i>r</i>)<i>m</i></span>. In particular, <i>S</i> itself is such an <i>R</i>-module.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Submodules_and_homomorphisms">Submodules and homomorphisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=5" title="Edit section: Submodules and homomorphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose <i>M</i> is a left <i>R</i>-module and <i>N</i> is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of <i>M</i>. Then <i>N</i> is a <b>submodule</b> (or more explicitly an <i>R</i>-submodule) if for any <i>n</i> in <i>N</i> and any <i>r</i> in <i>R</i>, the product <span class="nowrap"><i>r</i> ⋅ <i>n</i></span> (or <span class="nowrap"><i>n</i> ⋅ <i>r</i></span> for a right <i>R</i>-module) is in <i>N</i>. </p><p>If <i>X</i> is any <a href="/wiki/Subset" title="Subset">subset</a> of an <i>R</i>-module <i>M</i>, then the submodule spanned by <i>X</i> is defined to be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mspace width="thinmathspace" /> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>⊇</mo> <mi>X</mi> </mrow> </munder> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f579dff530e616643d4a18a802f99d2421b756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.032ex; height:3.176ex;" alt="{\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N}"></span> where <i>N</i> runs over the submodules of <i>M</i> that contain <i>X</i>, or explicitly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>X</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2584f3f85ad70bd4371292d114304e779ef08870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.194ex; height:4.843ex;" alt="{\textstyle \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}}"></span>, which is important in the definition of <a href="/wiki/Tensor_product_of_modules" title="Tensor product of modules">tensor products of modules</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The set of submodules of a given module <i>M</i>, together with the two binary operations + (the module spanned by the union of the arguments) and ∩, forms a <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> that satisfies the <b><a href="/wiki/Modular_lattice" title="Modular lattice">modular law</a></b>: Given submodules <i>U</i>, <i>N</i><sub>1</sub>, <i>N</i><sub>2</sub> of <i>M</i> such that <span class="nowrap"><i>N</i><sub>1</sub> ⊆ <i>N</i><sub>2</sub></span>, then the following two submodules are equal: <span class="nowrap">(<i>N</i><sub>1</sub> + <i>U</i>) ∩ <i>N</i><sub>2</sub> = <i>N</i><sub>1</sub> + (<i>U</i> ∩ <i>N</i><sub>2</sub>)</span>. </p><p>If <i>M</i> and <i>N</i> are left <i>R</i>-modules, then a <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">map</a> <span class="nowrap"><i>f</i> : <i>M</i> → <i>N</i></span> is a <b><a href="/wiki/Module_homomorphism" title="Module homomorphism">homomorphism of <i>R</i>-modules</a></b> if for any <i>m</i>, <i>n</i> in <i>M</i> and <i>r</i>, <i>s</i> in <i>R</i>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>⋅</mo> <mi>m</mi> <mo>+</mo> <mi>s</mi> <mo>⋅</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>s</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f36f8380e1c4a1342beadbdbe56964377ac42a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.908ex; height:2.843ex;" alt="{\displaystyle f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)}"></span>.</dd></dl> <p>This, like any <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> of mathematical objects, is just a mapping that preserves the structure of the objects. Another name for a homomorphism of <i>R</i>-modules is an <i>R</i>-<a href="/wiki/Linear_map" title="Linear map">linear map</a>. </p><p>A <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> module homomorphism <span class="nowrap"><i>f</i> : <i>M</i> → <i>N</i></span> is called a module <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>, and the two modules <i>M</i> and <i>N</i> are called <b>isomorphic</b>. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements. </p><p>The <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of a module homomorphism <span class="nowrap"><i>f</i> : <i>M</i> → <i>N</i></span> is the submodule of <i>M</i> consisting of all elements that are sent to zero by <i>f</i>, and the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of <i>f</i> is the submodule of <i>N</i> consisting of values <i>f</i>(<i>m</i>) for all elements <i>m</i> of <i>M</i>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Isomorphism_theorem" class="mw-redirect" title="Isomorphism theorem">isomorphism theorems</a> familiar from groups and vector spaces are also valid for <i>R</i>-modules. </p><p>Given a ring <i>R</i>, the set of all left <i>R</i>-modules together with their module homomorphisms forms an <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a>, denoted by <i>R</i>-<b>Mod</b> (see <a href="/wiki/Category_of_modules" title="Category of modules">category of modules</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_modules">Types of modules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=6" title="Edit section: Types of modules"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Glossary_of_module_theory" title="Glossary of module theory">Glossary of module theory</a></div> <dl><dt>Finitely generated</dt> <dd>An <i>R</i>-module <i>M</i> is <a href="/wiki/Finitely_generated_module" title="Finitely generated module">finitely generated</a> if there exist finitely many elements <i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub> in <i>M</i> such that every element of <i>M</i> is a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of those elements with coefficients from the ring <i>R</i>.</dd> <dt>Cyclic</dt> <dd>A module is called a <a href="/wiki/Cyclic_module" title="Cyclic module">cyclic module</a> if it is generated by one element.</dd> <dt>Free</dt> <dd>A <a href="/wiki/Free_module" title="Free module">free <i>R</i>-module</a> is a module that has a basis, or equivalently, one that is isomorphic to a <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> of copies of the ring <i>R</i>. These are the modules that behave very much like vector spaces.</dd> <dt>Projective</dt> <dd><a href="/wiki/Projective_module" title="Projective module">Projective modules</a> are <a href="/wiki/Direct_summand" class="mw-redirect" title="Direct summand">direct summands</a> of free modules and share many of their desirable properties.</dd> <dt>Injective</dt> <dd><a href="/wiki/Injective_module" title="Injective module">Injective modules</a> are defined dually to projective modules.</dd> <dt>Flat</dt> <dd>A module is called <a href="/wiki/Flat_module" title="Flat module">flat</a> if taking the <a href="/wiki/Tensor_product_of_modules" title="Tensor product of modules">tensor product</a> of it with any <a href="/wiki/Exact_sequence" title="Exact sequence">exact sequence</a> of <i>R</i>-modules preserves exactness.</dd> <dt>Torsionless</dt> <dd>A module is called <a href="/wiki/Torsionless_module" title="Torsionless module">torsionless</a> if it embeds into its <a href="/wiki/Dual_module" title="Dual module">algebraic dual</a>.</dd> <dt>Simple</dt> <dd>A <a href="/wiki/Simple_module" title="Simple module">simple module</a> <i>S</i> is a module that is not {0} and whose only submodules are {0} and <i>S</i>. Simple modules are sometimes called <i>irreducible</i>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></dd> <dt>Semisimple</dt> <dd>A <a href="/wiki/Semisimple_module" title="Semisimple module">semisimple module</a> is a direct sum (finite or not) of simple modules. Historically these modules are also called <i>completely reducible</i>.</dd> <dt>Indecomposable</dt> <dd>An <a href="/wiki/Indecomposable_module" title="Indecomposable module">indecomposable module</a> is a non-zero module that cannot be written as a <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules that are not simple (e.g. <a href="/wiki/Uniform_module" title="Uniform module">uniform modules</a>).</dd> <dt>Faithful</dt> <dd>A <a href="/wiki/Faithful_module" class="mw-redirect" title="Faithful module">faithful module</a> <i>M</i> is one where the action of each <span class="nowrap"><i>r</i> ≠ 0</span> in <i>R</i> on <i>M</i> is nontrivial (i.e. <span class="nowrap"><i>r</i> ⋅ <i>x</i> ≠ 0</span> for some <i>x</i> in <i>M</i>). Equivalently, the <a href="/wiki/Annihilator_(ring_theory)" title="Annihilator (ring theory)">annihilator</a> of <i>M</i> is the <a href="/wiki/Zero_ideal" class="mw-redirect" title="Zero ideal">zero ideal</a>.</dd> <dt>Torsion-free</dt> <dd>A <a href="/wiki/Torsion-free_module" title="Torsion-free module">torsion-free module</a> is a module over a ring such that 0 is the only element annihilated by a regular element (non <a href="/wiki/Zero-divisor" class="mw-redirect" title="Zero-divisor">zero-divisor</a>) of the ring, equivalently <span class="nowrap"><i>rm</i> = 0</span> implies <span class="nowrap"><i>r</i> = 0</span> or <span class="nowrap"><i>m</i> = 0</span>.</dd> <dt>Noetherian</dt> <dd>A <a href="/wiki/Noetherian_module" title="Noetherian module">Noetherian module</a> is a module that satisfies the <a href="/wiki/Ascending_chain_condition" title="Ascending chain condition">ascending chain condition</a> on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.</dd> <dt>Artinian</dt> <dd>An <a href="/wiki/Artinian_module" title="Artinian module">Artinian module</a> is a module that satisfies the <a href="/wiki/Descending_chain_condition" class="mw-redirect" title="Descending chain condition">descending chain condition</a> on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.</dd> <dt>Graded</dt> <dd>A <a href="/wiki/Graded_module" class="mw-redirect" title="Graded module">graded module</a> is a module with a decomposition as a direct sum <span class="nowrap"><i>M</i> = <span style="font-size:140%;">⨁</span><sub><i>x</i></sub> <i>M</i><sub><i>x</i></sub></span> over a <a href="/wiki/Graded_ring" title="Graded ring">graded ring</a> <span class="nowrap"><i>R</i> = <span style="font-size:140%;">⨁</span><sub><i>x</i></sub> <i>R</i><sub><i>x</i></sub></span> such that <span class="nowrap"><i>R</i><sub><i>x</i></sub><i>M</i><sub><i>y</i></sub> ⊆ <i>M</i><sub><i>x</i>+<i>y</i></sub></span> for all <i>x</i> and <i>y</i>.</dd> <dt>Uniform</dt> <dd>A <a href="/wiki/Uniform_module" title="Uniform module">uniform module</a> is a module in which all pairs of nonzero submodules have nonzero intersection.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Further_notions">Further notions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=7" title="Edit section: Further notions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Relation_to_representation_theory">Relation to representation theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=8" title="Edit section: Relation to representation theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A representation of a group <i>G</i> over a field <i>k</i> is a module over the <a href="/wiki/Group_ring" title="Group ring">group ring</a> <i>k</i>[<i>G</i>]. </p><p>If <i>M</i> is a left <i>R</i>-module, then the <i>action</i> of an element <i>r</i> in <i>R</i> is defined to be the map <span class="nowrap"><i>M</i> → <i>M</i></span> that sends each <i>x</i> to <i>rx</i> (or <i>xr</i> in the case of a right module), and is necessarily a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group endomorphism</a> of the abelian group <span class="nowrap">(<i>M</i>, +)</span>. The set of all group endomorphisms of <i>M</i> is denoted End<sub><b>Z</b></sub>(<i>M</i>) and forms a ring under addition and <a href="/wiki/Function_composition" title="Function composition">composition</a>, and sending a ring element <i>r</i> of <i>R</i> to its action actually defines a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> from <i>R</i> to End<sub><b>Z</b></sub>(<i>M</i>). </p><p>Such a ring homomorphism <span class="nowrap"><i>R</i> → End<sub><b>Z</b></sub>(<i>M</i>)</span> is called a <i>representation</i> of <i>R</i> over the abelian group <i>M</i>; an alternative and equivalent way of defining left <i>R</i>-modules is to say that a left <i>R</i>-module is an abelian group <i>M</i> together with a representation of <i>R</i> over it. Such a representation <span class="nowrap"><i>R</i> → End<sub><b>Z</b></sub>(<i>M</i>)</span> may also be called a <i>ring action</i> of <i>R</i> on <i>M</i>. </p><p>A representation is called <i>faithful</i> if and only if the map <span class="nowrap"><i>R</i> → End<sub><b>Z</b></sub>(<i>M</i>)</span> is <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a>. In terms of modules, this means that if <i>r</i> is an element of <i>R</i> such that <span class="nowrap"><i>rx</i> = 0</span> for all <i>x</i> in <i>M</i>, then <span class="nowrap"><i>r</i> = 0</span>. Every abelian group is a faithful module over the <a href="/wiki/Integer" title="Integer">integers</a> or over some <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">ring of integers modulo <i>n</i></a>, <b>Z</b>/<i>n</i><b>Z</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Generalizations">Generalizations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=9" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A ring <i>R</i> corresponds to a <a href="/wiki/Preadditive_category" title="Preadditive category">preadditive category</a> <b>R</b> with a single <a href="/wiki/Object_(category_theory)" class="mw-redirect" title="Object (category theory)">object</a>. With this understanding, a left <i>R</i>-module is just a covariant <a href="/wiki/Additive_functor" class="mw-redirect" title="Additive functor">additive functor</a> from <b>R</b> to the <a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">category <b>Ab</b> of abelian groups</a>, and right <i>R</i>-modules are contravariant additive functors. This suggests that, if <b>C</b> is any preadditive category, a covariant additive functor from <b>C</b> to <b>Ab</b> should be considered a generalized left module over <b>C</b>. These functors form a <a href="/wiki/Functor_category" title="Functor category">functor category</a> <b>C</b>-<b>Mod</b>, which is the natural generalization of the module category <i>R</i>-<b>Mod</b>. </p><p>Modules over <i>commutative</i> rings can be generalized in a different direction: take a <a href="/wiki/Ringed_space" title="Ringed space">ringed space</a> (<i>X</i>, O<sub><i>X</i></sub>) and consider the <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a> of O<sub><i>X</i></sub>-modules (see <a href="/wiki/Sheaf_of_modules" title="Sheaf of modules">sheaf of modules</a>). These form a category O<sub><i>X</i></sub>-<b>Mod</b>, and play an important role in modern <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. If <i>X</i> has only a single point, then this is a module category in the old sense over the commutative ring O<sub><i>X</i></sub>(<i>X</i>). </p><p>One can also consider modules over a <a href="/wiki/Semiring" title="Semiring">semiring</a>. Modules over rings are abelian groups, but modules over semirings are only <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="/wiki/Monoid" title="Monoid">monoids</a>. Most applications of modules are still possible. In particular, for any <a href="/wiki/Semiring" title="Semiring">semiring</a> <i>S</i>, the matrices over <i>S</i> form a semiring over which the tuples of elements from <i>S</i> are a module (in this generalized sense only). This allows a further generalization of the concept of <a href="/wiki/Vector_space" title="Vector space">vector space</a> incorporating the semirings from theoretical computer science. </p><p>Over <a href="/wiki/Near-rings" class="mw-redirect" title="Near-rings">near-rings</a>, one can consider near-ring modules, a nonabelian generalization of modules.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2015)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Group_ring" title="Group ring">Group ring</a></li> <li><a href="/wiki/Algebra_(ring_theory)" class="mw-redirect" title="Algebra (ring theory)">Algebra (ring theory)</a></li> <li><a href="/wiki/Module_(model_theory)" class="mw-redirect" title="Module (model theory)">Module (model theory)</a></li> <li><a href="/wiki/Module_spectrum" title="Module spectrum">Module spectrum</a></li> <li><a href="/wiki/Annihilator_(ring_theory)" title="Annihilator (ring theory)">Annihilator</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Hungerford (1974) <i>Algebra</i>, Springer, p 169: "Modules over a ring are a generalization of abelian groups (which are modules over Z)."</span> </li> <li id="cite_note-DummitFoote-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-DummitFoote_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDummit,_David_S.Foote,_Richard_M.2004" class="citation book cs1">Dummit, David S. & Foote, Richard M. (2004). <i>Abstract Algebra</i>. Hoboken, NJ: John Wiley & Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-43334-7" title="Special:BookSources/978-0-471-43334-7"><bdi>978-0-471-43334-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+Algebra&rft.place=Hoboken%2C+NJ&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=2004&rft.isbn=978-0-471-43334-7&rft.au=Dummit%2C+David+S.&rft.au=Foote%2C+Richard+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModule+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcgerty2016" class="citation web cs1">Mcgerty, Kevin (2016). <a rel="nofollow" class="external text" href="http://people.maths.ox.ac.uk/mcgerty/Algebra%20II.pdf">"ALGEBRA II: RINGS AND MODULES"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ALGEBRA+II%3A+RINGS+AND+MODULES&rft.date=2016&rft.aulast=Mcgerty&rft.aufirst=Kevin&rft_id=http%3A%2F%2Fpeople.maths.ox.ac.uk%2Fmcgerty%2FAlgebra%2520II.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModule+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAsh" class="citation web cs1">Ash, Robert. <a rel="nofollow" class="external text" href="https://faculty.math.illinois.edu/~r-ash/Algebra/Chapter4.pdf">"Module Fundamentals"</a> <span class="cs1-format">(PDF)</span>. <i>Abstract Algebra: The Basic Graduate Year</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Abstract+Algebra%3A+The+Basic+Graduate+Year&rft.atitle=Module+Fundamentals&rft.aulast=Ash&rft.aufirst=Robert&rft_id=https%3A%2F%2Ffaculty.math.illinois.edu%2F~r-ash%2FAlgebra%2FChapter4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModule+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Jacobson (1964), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KlMDjaJxZAkC&pg=PA4">p. 4</a>, Def. 1</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>F.W. Anderson and K.R. Fuller: <i>Rings and Categories of Modules</i>, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97845-3" title="Special:BookSources/0-387-97845-3">0-387-97845-3</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-97845-3" title="Special:BookSources/3-540-97845-3">3-540-97845-3</a></li> <li><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Nathan Jacobson</a>. <i>Structure of rings</i>. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1037-8" title="Special:BookSources/978-0-8218-1037-8">978-0-8218-1037-8</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Module_(mathematics)&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Module">"Module"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Module&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DModule&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModule+%28mathematics%29" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/module">module</a> at the <a href="/wiki/NLab" title="NLab"><i>n</i>Lab</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q18848#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q18848#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q18848#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="http://id.worldcat.org/fast/1024523/">FAST</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85086470">United States</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb13163015r">France</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb13163015r">BnF data</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00564457">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="moduly (algebra)"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph211266&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX526925">Spain</a></span></li><li><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=987007541015705171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.idref.fr/027814572">IdRef</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Module_(mathematics)&oldid=1251812964">https://en.wikipedia.org/w/index.php?title=Module_(mathematics)&oldid=1251812964</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Algebraic_structures" title="Category:Algebraic structures">Algebraic structures</a></li><li><a href="/wiki/Category:Module_theory" title="Category:Module theory">Module theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_May_2015" title="Category:Articles lacking in-text citations from May 2015">Articles lacking in-text citations from May 2015</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2015" title="Category:Articles with unsourced statements from May 2015">Articles with unsourced statements from May 2015</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 18 October 2024, at 06:20<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Module_(mathematics)&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-v8npv","wgBackendResponseTime":160,"wgPageParseReport":{"limitreport":{"cputime":"0.572","walltime":"0.779","ppvisitednodes":{"value":2431,"limit":1000000},"postexpandincludesize":{"value":50609,"limit":2097152},"templateargumentsize":{"value":3454,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":4,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":71525,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 614.802 1 -total"," 22.90% 140.810 1 Template:Reflist"," 18.16% 111.636 2 Template:Sidebar_with_collapsible_lists"," 17.95% 110.378 1 Template:Cite_book"," 16.36% 100.571 1 Template:Ring_theory_sidebar"," 12.81% 78.783 1 Template:Authority_control"," 10.79% 66.346 1 Template:Short_description"," 9.89% 60.785 1 Template:More_footnotes"," 8.64% 53.106 1 Template:Ambox"," 6.82% 41.929 2 Template:Pagetype"]},"scribunto":{"limitreport-timeusage":{"value":"0.348","limit":"10.000"},"limitreport-memusage":{"value":5525180,"limit":52428800}},"cachereport":{"origin":"mw-api-int.codfw.main-664bdf464b-lbvxg","timestamp":"20241128172527","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Module (mathematics)","url":"https:\/\/en.wikipedia.org\/wiki\/Module_(mathematics)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q18848","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q18848","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-07-24T15:25:11Z","dateModified":"2024-10-18T06:20:39Z","headline":"generalization of vector space, with scalars in a ring instead of a field"}</script> </body> </html>