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<a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Voir aussi</span> </div> </a> <ul id="toc-Voir_aussi-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="Sommaire" 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="Basculer la table des matières" > <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">Basculer la table des matières</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">Élément neutre</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="Aller à un article dans une autre langue. Disponible en 49 langues." > <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-49" 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">49 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D9%86%D8%B5%D8%B1_%D9%85%D8%AD%D8%A7%D9%8A%D8%AF" title="عنصر محايد – arabe" lang="ar" hreflang="ar" data-title="عنصر محايد" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Elementu_neutru" title="Elementu neutru – asturien" lang="ast" hreflang="ast" data-title="Elementu neutru" data-language-autonym="Asturianu" data-language-local-name="asturien" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%B5%D1%83%D1%82%D1%80%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B5%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82" title="Неутрален елемент – bulgare" lang="bg" hreflang="bg" data-title="Неутрален елемент" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Neutralni_element" title="Neutralni element – bosniaque" lang="bs" hreflang="bs" data-title="Neutralni element" data-language-autonym="Bosanski" data-language-local-name="bosniaque" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Element_neutre" title="Element neutre – catalan" lang="ca" hreflang="ca" data-title="Element neutre" 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/Neutr%C3%A1ln%C3%AD_prvek" title="Neutrální prvek – tchèque" lang="cs" hreflang="cs" data-title="Neutrální prvek" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Neutralt_element" title="Neutralt element – danois" lang="da" hreflang="da" data-title="Neutralt element" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Neutrales_Element" title="Neutrales Element – allemand" lang="de" hreflang="de" data-title="Neutrales Element" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CF%85%CE%B4%CE%AD%CF%84%CE%B5%CF%81%CE%BF_%CF%83%CF%84%CE%BF%CE%B9%CF%87%CE%B5%CE%AF%CE%BF" title="Ουδέτερο στοιχείο – grec" lang="el" hreflang="el" data-title="Ουδέτερο στοιχείο" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Identity_element" title="Identity element – anglais" lang="en" hreflang="en" data-title="Identity element" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ne%C5%ADtrala_elemento_(matematiko)" title="Neŭtrala elemento (matematiko) – espéranto" lang="eo" hreflang="eo" data-title="Neŭtrala elemento (matematiko)" data-language-autonym="Esperanto" data-language-local-name="espéranto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Elemento_neutro" title="Elemento neutro – espagnol" lang="es" hreflang="es" data-title="Elemento neutro" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/%C3%9Chikelement" title="Ühikelement – estonien" lang="et" hreflang="et" data-title="Ühikelement" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Elementu_neutro" title="Elementu neutro – basque" lang="eu" hreflang="eu" data-title="Elementu neutro" 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/%D8%B9%D9%85%D9%84_%D8%AF%D9%88%D8%AA%D8%A7%DB%8C%DB%8C#عضو_خنثی" title="عمل دوتایی – persan" lang="fa" hreflang="fa" data-title="عمل دوتایی" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Neutraalialkio" title="Neutraalialkio – finnois" lang="fi" hreflang="fi" data-title="Neutraalialkio" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Neutraal_element" title="Neutraal element – frison septentrional" lang="frr" hreflang="frr" data-title="Neutraal element" data-language-autonym="Nordfriisk" data-language-local-name="frison septentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Elemento_neutro" title="Elemento neutro – galicien" lang="gl" hreflang="gl" data-title="Elemento neutro" data-language-autonym="Galego" data-language-local-name="galicien" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%91%D7%A8_%D7%99%D7%97%D7%99%D7%93%D7%94" title="איבר יחידה – hébreu" lang="he" hreflang="he" data-title="איבר יחידה" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Neutralni_element" title="Neutralni element – croate" lang="hr" hreflang="hr" data-title="Neutralni element" data-language-autonym="Hrvatski" data-language-local-name="croate" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Neutr%C3%A1lis_elem" title="Neutrális elem – hongrois" lang="hu" hreflang="hu" data-title="Neutrális elem" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Elemento_neutre" title="Elemento neutre – interlingua" lang="ia" hreflang="ia" data-title="Elemento neutre" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Unsur_identitas" title="Unsur identitas – indonésien" lang="id" hreflang="id" data-title="Unsur identitas" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hlutleysa" title="Hlutleysa – islandais" lang="is" hreflang="is" data-title="Hlutleysa" data-language-autonym="Íslenska" data-language-local-name="islandais" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Elemento_neutro" title="Elemento neutro – italien" lang="it" hreflang="it" data-title="Elemento neutro" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8D%98%E4%BD%8D%E5%85%83" title="単位元 – japonais" lang="ja" hreflang="ja" data-title="単位元" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%AD%EB%93%B1%EC%9B%90" title="항등원 – coréen" lang="ko" hreflang="ko" data-title="항등원" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Element_neuter" title="Element neuter – lombard" lang="lmo" hreflang="lmo" data-title="Element neuter" data-language-autonym="Lombard" data-language-local-name="lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%BD%E0%B4%B8%E0%B4%AE%E0%B4%95%E0%B4%82" title="തൽസമകം – malayalam" lang="ml" hreflang="ml" data-title="തൽസമകം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Neutraal_element" title="Neutraal element – néerlandais" lang="nl" hreflang="nl" data-title="Neutraal element" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Identitetselement" title="Identitetselement – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Identitetselement" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Element_neutralny" title="Element neutralny – polonais" lang="pl" hreflang="pl" data-title="Element neutralny" data-language-autonym="Polski" data-language-local-name="polonais" 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/Elemento_neutro" title="Elemento neutro – portugais" lang="pt" hreflang="pt" data-title="Elemento neutro" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Element_neutru" title="Element neutru – roumain" lang="ro" hreflang="ro" data-title="Element neutru" data-language-autonym="Română" data-language-local-name="roumain" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B9%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82" title="Нейтральный элемент – russe" lang="ru" hreflang="ru" data-title="Нейтральный элемент" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Neutral_(matematika)" title="Neutral (matematika) – serbo-croate" lang="sh" hreflang="sh" data-title="Neutral (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croate" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Identity_element" title="Identity element – Simple English" lang="en-simple" hreflang="en-simple" data-title="Identity element" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Neutr%C3%A1lny_prvok" title="Neutrálny prvok – slovaque" lang="sk" hreflang="sk" data-title="Neutrálny prvok" data-language-autonym="Slovenčina" data-language-local-name="slovaque" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Nevtralni_element" title="Nevtralni element – slovène" lang="sl" hreflang="sl" data-title="Nevtralni element" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%B5%D1%83%D1%82%D1%80%D0%B0%D0%BB" title="Неутрал – serbe" lang="sr" hreflang="sr" data-title="Неутрал" data-language-autonym="Српски / srpski" data-language-local-name="serbe" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Neutralt_element" title="Neutralt element – suédois" lang="sv" hreflang="sv" data-title="Neutralt element" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%8A%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AF%88_%E0%AE%89%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="முற்றொருமை உறுப்பு – tamoul" lang="ta" hreflang="ta" data-title="முற்றொருமை உறுப்பு" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%B2%E0%B8%8A%E0%B8%B4%E0%B8%81%E0%B9%80%E0%B8%AD%E0%B8%81%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%A9%E0%B8%93%E0%B9%8C" title="สมาชิกเอกลักษณ์ – thaï" lang="th" hreflang="th" data-title="สมาชิกเอกลักษณ์" data-language-autonym="ไทย" data-language-local-name="thaï" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Birim_%C3%B6ge" title="Birim öge – turc" lang="tr" hreflang="tr" data-title="Birim öge" data-language-autonym="Türkçe" data-language-local-name="turc" 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%9D%D0%B5%D0%B9%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%B5%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82" title="Нейтральний елемент – ukrainien" lang="uk" hreflang="uk" data-title="Нейтральний елемент" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%E1%BA%A7n_t%E1%BB%AD_%C4%91%C6%A1n_v%E1%BB%8B" title="Phần tử đơn vị – vietnamien" lang="vi" hreflang="vi" data-title="Phần tử đơn vị" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A0%D7%99%D7%99%D7%98%D7%A8%D7%90%D7%9C%D7%A2%D7%A8_%D7%A2%D7%9C%D7%A2%D7%9E%D7%A2%D7%A0%D7%98" title="נייטראלער עלעמענט – yiddish" lang="yi" hreflang="yi" data-title="נייטראלער עלעמענט" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%96%AE%E4%BD%8D%E5%85%83" title="單位元 – chinois" lang="zh" hreflang="zh" data-title="單位元" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%96%AE%E4%BD%8D%E5%85%83" title="單位元 – cantonais" lang="yue" hreflang="yue" data-title="單位元" data-language-autonym="粵語" data-language-local-name="cantonais" class="interlanguage-link-target"><span>粵語</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/Q185813#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</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="Espaces de noms"> <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 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class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <a href="/wiki/%C3%89l%C3%A9ment" class="mw-disambig" title="Élément">Élément</a> et <a href="/wiki/Neutre" class="mw-disambig" title="Neutre">Neutre</a>. </p> </div></div> <p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, plus précisément en <a href="/wiki/Alg%C3%A8bre" title="Algèbre">algèbre</a>, un <b>élément neutre</b> (ou <b>élément identité</b>) d'un <a href="/wiki/Ensemble" title="Ensemble">ensemble</a> pour une <a href="/wiki/Loi_de_composition_interne" title="Loi de composition interne">loi de composition interne</a> est un élément de cet ensemble qui laisse tous les autres éléments inchangés lorsqu'il est composé avec eux par cette loi. Un <a href="/wiki/Magma_(alg%C3%A8bre)" title="Magma (algèbre)">magma</a> possédant un élément neutre est dit <b>unifère</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définition"><span id="D.C3.A9finition"></span>Définition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&veaction=edit&section=1" title="Modifier la section : Définition" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&action=edit&section=1" title="Modifier le code source de la section : Définition"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Soit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\star )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mo>⋆</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\star )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10212ead787fda4b45e489b476439f24562389e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:2.843ex;" alt="{\displaystyle (E,\star )}"></span> un magma. Un élément <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> est dit : </p> <ul><li><i>neutre à gauche</i> si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in E,\ e\star x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mtext> </mtext> <mi>e</mi> <mo>⋆</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in E,\ e\star x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4ad7e73f89a2b26f51d3472f6e49850629b91d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.889ex; height:2.509ex;" alt="{\displaystyle \forall x\in E,\ e\star x=x}"></span> ;</li> <li><i>neutre à droite</i> si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in E,\ x\star e=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mtext> </mtext> <mi>x</mi> <mo>⋆</mo> <mi>e</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in E,\ x\star e=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad77afb26b2868a59de0f7077e1c7241cbb42cf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.889ex; height:2.509ex;" alt="{\displaystyle \forall x\in E,\ x\star e=x}"></span> ;</li> <li><i>neutre</i> s'il est neutre à droite et à gauche.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Exemples">Exemples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&veaction=edit&section=2" title="Modifier la section : Exemples" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&action=edit&section=2" title="Modifier le code source de la section : Exemples"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un élément neutre est relatif à la loi considérée : </p> <ul><li>0 est l'élément neutre de l’<a href="/wiki/Addition" title="Addition">addition</a> arithmétique, ainsi que du « <a href="/wiki/Disjonction_logique" title="Disjonction logique">ou</a> » binaire ;</li> <li>1 est l'élément neutre de la <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> arithmétique, ainsi que du « <a href="/wiki/Fonction_ET" title="Fonction ET">et</a> » binaire ;</li> <li>La <a href="/wiki/Matrice_unit%C3%A9" class="mw-redirect" title="Matrice unité">matrice unité</a> d'ordre <i>n</i> est l'élément neutre de la <a href="/wiki/Multiplication_des_matrices" class="mw-redirect" title="Multiplication des matrices">multiplication</a> des <a href="/wiki/Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques)">matrices</a> carrées d'ordre <i>n</i> ;</li> <li>Le <a href="/wiki/Mot_(math%C3%A9matiques)" title="Mot (mathématiques)">mot</a> vide est l'élément neutre de la <a href="/wiki/Concat%C3%A9nation" title="Concaténation">concaténation</a> des chaînes de caractères.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Propriétés"><span id="Propri.C3.A9t.C3.A9s"></span>Propriétés</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&veaction=edit&section=3" title="Modifier la section : Propriétés" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&action=edit&section=3" title="Modifier le code source de la section : Propriétés"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Il est possible que l'élément neutre à gauche (resp. à droite) ne soit pas unique. Par exemple, considérons un ensemble <i>E</i> contenant au moins deux éléments. On peut définir une loi <i>G</i> sur <i>E</i> par la formule <i>G</i>(<i>x</i>, <i>y</i>) = <i>x</i> et une loi <i>D</i> par la formule <i>D</i>(<i>x</i>, <i>y</i>) = <i>y</i>. Pour la loi <i>G</i>, tout élément est neutre à droite et aucun n'est neutre à gauche. Pour la loi <i>D</i>, tout élément est neutre à gauche et aucun n'est neutre à droite.</li> <li>En revanche, s'il existe un élément neutre à gauche et un élément neutre à droite, alors l'ensemble admet un unique élément neutre et en outre, tout élément neutre à gauche (resp. à droite) lui est égal. En effet, pour tous éléments <i>e<sub>g</sub></i> neutre à gauche et <i>e<sub>d</sub></i> neutre à droite, on a : <i>e<sub>d</sub> = e<sub>g</sub>e<sub>d</sub> = e<sub>g</sub></i>.<br />Dans cette situation — en particulier lorsque <i>G</i> est un <a href="/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup> —, l'unique élément neutre de <i>G</i> est couramment appelé <b>le neutre</b> de <i>G</i>.</li> <li>Dans un <a href="/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</a>, le neutre est le seul élément <a href="/wiki/Idempotence" title="Idempotence">idempotent</a>, c'est-à-dire le seul élément <i>x</i> tel que <i>x x = x</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&veaction=edit&section=4" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&action=edit&section=4" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><a href="/wiki/N._Bourbaki" class="mw-redirect" title="N. Bourbaki">N. Bourbaki</a>, <i><a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Algèbre</a></i>, chap. I, <abbr class="abbr" title="page(s)">p.</abbr> I-12.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Szpirglas"><span class="ouvrage" id="Aviva_Szpirglas">Aviva <span class="nom_auteur">Szpirglas</span>, <cite class="italique">Algèbre L3 : Cours complet avec 400 tests et exercices corrigés</cite> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Alg%C3%A8bre_L3_(Szpirglas)" title="Référence:Algèbre L3 (Szpirglas)">détail de l’édition</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alg%C3%A8bre+L3+%3A+Cours+complet+avec+400+tests+et+exercices+corrig%C3%A9s&rft.aulast=Szpirglas&rft.aufirst=Aviva&rfr_id=info%3Asid%2Ffr.wikipedia.org%3A%C3%89l%C3%A9ment+neutre"></span></span></span>, Partie III : Groupes, Exemples 6.2.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text">Jean-Pierre Marco et Laurent Lazzarini, <i>Mathématiques L1</i>, <abbr class="abbr" title="page">p.</abbr> 119, <span class="noarchive"><a rel="nofollow" class="external text" href="//books.google.com/books?id=r1bzeQoVx_MC&pg=PA119"><i>aperçu</i></a></span> sur <i><a href="/wiki/Google_Livres" title="Google Livres">Google Livres</a></i>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink noprint"><a href="#cite_ref-4">↑</a> </span><span class="reference-text">Jean-Marie Monier, <i>Méthodes et Exercices de Mathématiques MPSI</i>, <abbr class="abbr" title="page">p.</abbr> 213, <span class="noarchive"><a rel="nofollow" class="external text" href="//books.google.com/books?id=TKhld15V1jcC&pg=PA213"><i>aperçu</i></a></span> sur <i><a href="/wiki/Google_Livres" title="Google Livres">Google Livres</a></i>.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&veaction=edit&section=5" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89l%C3%A9ment_neutre&action=edit&section=5" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="column-count:3;column-gap:1em;" class="colonnes"> <ul><li><a href="/wiki/Anneau_unif%C3%A8re" class="mw-redirect" title="Anneau unifère">Anneau unifère</a></li> <li><a href="/wiki/%C3%89l%C3%A9ment_absorbant" title="Élément absorbant">Élément absorbant</a></li> <li><a href="/wiki/%C3%89l%C3%A9ment_sym%C3%A9trique" title="Élément symétrique">Élément symétrique</a></li> <li><a href="/wiki/Inverse" title="Inverse">Inverse</a></li> <li><a href="/wiki/Mono%C3%AFde" title="Monoïde">Monoïde</a></li> <li><a href="/wiki/Quasigroupe" title="Quasigroupe">Quasigroupe</a></li></ul> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Alg%C3%A8bre" title="Portail de l’algèbre"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/24px-Arithmetic_symbols.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/36px-Arithmetic_symbols.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/48px-Arithmetic_symbols.svg.png 2x" data-file-width="210" data-file-height="210" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Alg%C3%A8bre" title="Portail:Algèbre">Portail de l’algèbre</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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