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mw-first-heading"><span class="mw-page-title-main">Identity element</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 49 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-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 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/%D8%B9%D9%86%D8%B5%D8%B1_%D9%85%D8%AD%D8%A7%D9%8A%D8%AF" title="عنصر محايد – Arabic" lang="ar" hreflang="ar" data-title="عنصر محايد" data-language-autonym="العربية" data-language-local-name="Arabic" 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 – Asturian" lang="ast" hreflang="ast" data-title="Elementu neutru" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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 – Bulgarian" lang="bg" hreflang="bg" data-title="Неутрален елемент" data-language-autonym="Български" data-language-local-name="Bulgarian" 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 – Bosnian" lang="bs" hreflang="bs" data-title="Neutralni element" data-language-autonym="Bosanski" data-language-local-name="Bosnian" 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 – Czech" lang="cs" hreflang="cs" data-title="Neutrální prvek" data-language-autonym="Čeština" data-language-local-name="Czech" 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 – Danish" lang="da" hreflang="da" data-title="Neutralt element" data-language-autonym="Dansk" data-language-local-name="Danish" 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 – German" lang="de" hreflang="de" data-title="Neutrales Element" 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/%C3%9Chikelement" title="Ühikelement – Estonian" lang="et" hreflang="et" data-title="Ühikelement" 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%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 – Greek" lang="el" hreflang="el" data-title="Ουδέτερο στοιχείο" 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/Elemento_neutro" title="Elemento neutro – Spanish" lang="es" hreflang="es" data-title="Elemento neutro" 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/Ne%C5%ADtrala_elemento_(matematiko)" title="Neŭtrala elemento (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Neŭtrala elemento (matematiko)" 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/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%86%D8%B5%D8%B1_%D9%87%D9%85%D8%A7%D9%86%DB%8C" title="عنصر همانی – Persian" lang="fa" hreflang="fa" data-title="عنصر همانی" 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/%C3%89l%C3%A9ment_neutre" title="Élément neutre – French" lang="fr" hreflang="fr" data-title="Élément neutre" 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/Elemento_neutro" title="Elemento neutro – Galician" lang="gl" hreflang="gl" data-title="Elemento neutro" 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/%ED%95%AD%EB%93%B1%EC%9B%90" title="항등원 – Korean" lang="ko" hreflang="ko" data-title="항등원" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Neutralni_element" title="Neutralni element – Croatian" lang="hr" hreflang="hr" data-title="Neutralni element" 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/Unsur_identitas" title="Unsur identitas – Indonesian" lang="id" hreflang="id" data-title="Unsur identitas" 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/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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hlutleysa" title="Hlutleysa – Icelandic" lang="is" hreflang="is" data-title="Hlutleysa" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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 – Italian" lang="it" hreflang="it" data-title="Elemento neutro" 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%90%D7%99%D7%91%D7%A8_%D7%99%D7%97%D7%99%D7%93%D7%94" title="איבר יחידה – Hebrew" lang="he" hreflang="he" data-title="איבר יחידה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</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-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Neutr%C3%A1lis_elem" title="Neutrális elem – Hungarian" lang="hu" hreflang="hu" data-title="Neutrális elem" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Dutch" lang="nl" hreflang="nl" data-title="Neutraal element" 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/%E5%8D%98%E4%BD%8D%E5%85%83" title="単位元 – Japanese" lang="ja" hreflang="ja" data-title="単位元" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Neutraal_element" title="Neutraal element – Northern Frisian" lang="frr" hreflang="frr" data-title="Neutraal element" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Identitetselement" title="Identitetselement – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Identitetselement" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Element_neutralny" title="Element neutralny – Polish" lang="pl" hreflang="pl" data-title="Element neutralny" 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/Elemento_neutro" title="Elemento neutro – Portuguese" lang="pt" hreflang="pt" data-title="Elemento neutro" data-language-autonym="Português" data-language-local-name="Portuguese" 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 – Romanian" lang="ro" hreflang="ro" data-title="Element neutru" data-language-autonym="Română" data-language-local-name="Romanian" 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 – Russian" lang="ru" hreflang="ru" data-title="Нейтральный элемент" 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/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 – Slovak" lang="sk" hreflang="sk" data-title="Neutrálny prvok" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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 – Slovenian" lang="sl" hreflang="sl" data-title="Nevtralni element" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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 – Serbian" lang="sr" hreflang="sr" data-title="Неутрал" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Neutral_(matematika)" title="Neutral (matematika) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Neutral (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Neutraalialkio" title="Neutraalialkio – Finnish" lang="fi" hreflang="fi" data-title="Neutraalialkio" 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/Neutralt_element" title="Neutralt element – Swedish" lang="sv" hreflang="sv" data-title="Neutralt element" data-language-autonym="Svenska" data-language-local-name="Swedish" 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 – Tamil" lang="ta" hreflang="ta" data-title="முற்றொருமை உறுப்பு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" 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 – Thai" lang="th" hreflang="th" data-title="สมาชิกเอกลักษณ์" data-language-autonym="ไทย" data-language-local-name="Thai" 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 – Turkish" lang="tr" hreflang="tr" data-title="Birim öge" 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%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 – Ukrainian" lang="uk" hreflang="uk" data-title="Нейтральний елемент" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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 – Vietnamese" lang="vi" hreflang="vi" data-title="Phần tử đơn vị" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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-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 – Cantonese" lang="yue" hreflang="yue" data-title="單位元" 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/%E5%96%AE%E4%BD%8D%E5%85%83" title="單位元 – Chinese" lang="zh" 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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">Specific element of an algebraic structure</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>identity element</b> or <b>neutral element</b> of a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> is an element that leaves unchanged every element when the operation is applied.<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For example, 0 is an identity element of the <a href="/wiki/Addition" title="Addition">addition</a> of <a href="/wiki/Real_number" title="Real number">real numbers</a>. This concept is used in <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a> such as <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> and <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>. The term <i>identity element</i> is often shortened to <i>identity</i> (as in the case of additive identity and multiplicative identity)<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_element&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml">(<i>S</i>, ∗)</span> be a set <span class="texhtml mvar" style="font-style:italic;">S</span> equipped with a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> ∗. Then an element <span class="texhtml mvar" style="font-style:italic;">e</span> of <span class="texhtml mvar" style="font-style:italic;">S</span> is called a <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="left_identity_element"></span><span class="vanchor-text"><b><a href="/wiki/Left_and_right_(algebra)" title="Left and right (algebra)">left</a> identity</b></span></span> if <span class="texhtml"><i>e</i> ∗ <i>s</i> = <i>s</i></span> for all <span class="texhtml mvar" style="font-style:italic;">s</span> in <span class="texhtml mvar" style="font-style:italic;">S</span>, and a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="right_identity_element"></span><span class="vanchor-text"><b><a href="/wiki/Left_and_right_(algebra)" title="Left and right (algebra)">right</a> identity</b></span></span> if <span class="texhtml"><i>s</i> ∗ <i>e</i> = <i>s</i></span> for all <span class="texhtml mvar" style="font-style:italic;">s</span> in <span class="texhtml mvar" style="font-style:italic;">S</span>.<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> If <span class="texhtml mvar" style="font-style:italic;">e</span> is both a left identity and a right identity, then it is called a <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="two-sided_identity"></span><span class="vanchor-text">two-sided identity</span></span></b>, or simply an <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="identity"></span><span class="vanchor-text">identity</span></span></b>.<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><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>An identity with respect to addition is called an <a href="/wiki/Additive_identity" title="Additive identity"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="additive_identity"></span><span class="vanchor-text">additive identity</span></span></a> (often denoted as 0) and an identity with respect to multiplication is called a <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="multiplicative_identity"></span><span class="vanchor-text">multiplicative identity</span></span></b> (often denoted as 1).<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> for example, the identity element is sometimes simply denoted by the symbol <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>. The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, <a href="/wiki/Integral_domain" title="Integral domain">integral domains</a>, and <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>. The multiplicative identity is often called <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="unity"></span><span class="vanchor-text">unity</span></span></b> in the latter context (a ring with unity).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This should not be confused with a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a> in ring theory, which is any element having a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>. By its own definition, unity itself is necessarily a unit.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </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=Identity_element&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th>Set</th> <th>Operation</th> <th>Identity </th></tr> <tr> <td rowspan="2"><a href="/wiki/Real_number" title="Real number">Real numbers</a>, <a href="/wiki/Complex_number" title="Complex number">Complex numbers</a>,<br /><a href="/wiki/Quaternion" title="Quaternion">Quaternions</a>, <a href="/wiki/Multivector" title="Multivector">Multivectors</a>, etc.</td> <td>+ (<a href="/wiki/Addition" title="Addition">addition</a>)</td> <td><a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0</a> </td></tr> <tr> <td>· (<a href="/wiki/Multiplication" title="Multiplication">multiplication</a>)</td> <td><a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a> </td></tr> <tr> <td><a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">Positive integers</a></td> <td><a href="/wiki/Least_common_multiple" title="Least common multiple">Least common multiple</a></td> <td>1 </td></tr> <tr> <td><a href="/wiki/Non-negative_integer" class="mw-redirect" title="Non-negative integer">Non-negative integers</a></td> <td><a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">Greatest common divisor</a></td> <td>0 (under most definitions of GCD) </td></tr> <tr> <td><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vectors</a></td> <td><a href="/wiki/Vector_addition" class="mw-redirect" title="Vector addition">Vector addition</a> </td> <td><a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">Zero vector</a> </td></tr> <tr> <td><span class="texhtml mvar" style="font-style:italic;">m</span>-by-<span class="texhtml mvar" style="font-style:italic;">n</span> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a></td> <td><a href="/wiki/Matrix_addition" title="Matrix addition">Matrix addition</a> </td> <td><a href="/wiki/Zero_matrix" title="Zero matrix">Zero matrix</a> </td></tr> <tr> <td><span class="texhtml mvar" style="font-style:italic;">n</span>-by-<span class="texhtml mvar" style="font-style:italic;">n</span> square matrices</td> <td><a href="/wiki/Matrix_multiplication" title="Matrix multiplication">Matrix multiplication</a> </td> <td><i>I</i><sub><i>n</i></sub> (<a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>) </td></tr> <tr> <td><span class="texhtml mvar" style="font-style:italic;">m</span>-by-<span class="texhtml mvar" style="font-style:italic;">n</span> matrices</td> <td>○ (<a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a>) </td> <td><span class="texhtml"><i>J</i><sub><i>m</i>, <i>n</i></sub></span> (<a href="/wiki/Matrix_of_ones" title="Matrix of ones">matrix of ones</a>) </td></tr> <tr> <td>All <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> from a set, <span class="texhtml mvar" style="font-style:italic;">M</span>, to itself</td> <td>∘ (<a href="/wiki/Function_composition" title="Function composition">function composition</a>)</td> <td><a href="/wiki/Identity_function" title="Identity function">Identity function</a> </td></tr> <tr> <td>All <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a> on a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, <span class="texhtml mvar" style="font-style:italic;">G</span></td> <td>∗ (<a href="/wiki/Convolution" title="Convolution">convolution</a>)</td> <td><span class="texhtml"><i>δ</i></span> (<a href="/wiki/Dirac_delta" class="mw-redirect" title="Dirac delta">Dirac delta</a>) </td></tr> <tr> <td rowspan="2"><a href="/wiki/Extended_real_number" class="mw-redirect" title="Extended real number">Extended real numbers</a></td> <td><a href="/wiki/Minimum" class="mw-redirect" title="Minimum">Minimum</a>/infimum</td> <td>+∞ </td></tr> <tr> <td><a href="/wiki/Maximum" class="mw-redirect" title="Maximum">Maximum</a>/supremum</td> <td>−∞ </td></tr> <tr> <td rowspan="2">Subsets of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml mvar" style="font-style:italic;">M</span></td> <td>∩ (<a href="/wiki/Set_intersection" class="mw-redirect" title="Set intersection">intersection</a>)</td> <td><span class="texhtml mvar" style="font-style:italic;">M</span> </td></tr> <tr> <td>∪ (<a href="/wiki/Set_union" class="mw-redirect" title="Set union">union</a>)</td> <td>∅ (<a href="/wiki/Empty_set" title="Empty set">empty set</a>) </td></tr> <tr> <td><a href="/wiki/String_(computer_science)" title="String (computer science)">Strings</a>, <a href="/wiki/Tuple" title="Tuple">lists</a></td> <td><a href="/wiki/Concatenation" title="Concatenation">Concatenation</a></td> <td><a href="/wiki/Empty_string" title="Empty string">Empty string</a>, empty list </td></tr> <tr> <td rowspan="4">A <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></td> <td>∧ (<a href="/wiki/Logical_and" class="mw-redirect" title="Logical and">logical and</a>)</td> <td>⊤ (truth) </td></tr> <tr> <td>↔ (<a href="/wiki/Logical_biconditional" title="Logical biconditional">logical biconditional</a>)</td> <td>⊤ (truth) </td></tr> <tr> <td>∨ (<a href="/wiki/Logical_or" class="mw-redirect" title="Logical or">logical or</a>)</td> <td>⊥ (falsity) </td></tr> <tr> <td>⊕ (<a href="/wiki/Exclusive_or" title="Exclusive or">exclusive or</a>)</td> <td>⊥ (falsity) </td></tr> <tr> <td><a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">Knots</a></td> <td><a href="/wiki/Knot_sum" class="mw-redirect" title="Knot sum">Knot sum</a></td> <td><a href="/wiki/Unknot" title="Unknot">Unknot</a> </td></tr> <tr> <td><a href="/wiki/Compact_surfaces" class="mw-redirect" title="Compact surfaces">Compact surfaces</a></td> <td># (<a href="/wiki/Connected_sum" title="Connected sum">connected sum</a>)</td> <td><a href="/wiki/Sphere" title="Sphere"><i>S</i><sup>2</sup></a> </td></tr> <tr> <td><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Groups</a></td> <td><a href="/wiki/Direct_product" title="Direct product">Direct product</a></td> <td><a href="/wiki/Trivial_group" title="Trivial group">Trivial group</a> </td></tr> <tr> <td>Two elements, <span class="texhtml">{<i>e</i>, <i>f</i>} </span> </td> <td>∗ defined by<br /> <span class="texhtml"><i>e</i> ∗ <i>e</i> = <i>f</i> ∗ <i>e</i> = <i>e</i></span> and <br /> <span class="texhtml"><i>f</i> ∗ <i>f</i> = <i>e</i> ∗ <i>f</i> = <i>f</i></span> </td> <td>Both <span class="texhtml mvar" style="font-style:italic;">e</span> and <span class="texhtml mvar" style="font-style:italic;">f</span> are left identities,<br /> but there is no right identity<br /> and no two-sided identity </td></tr> <tr> <td><a href="/wiki/Homogeneous_relation" title="Homogeneous relation">Homogeneous relations</a> on a set <i>X</i></td> <td><a href="/wiki/Relative_product" class="mw-redirect" title="Relative product">Relative product</a></td> <td><a href="/wiki/Identity_relation" class="mw-redirect" title="Identity relation">Identity relation</a> </td></tr> <tr> <td><a href="/wiki/Relational_algebra" title="Relational algebra">Relational algebra</a></td> <td><a href="/wiki/Natural_join" class="mw-redirect" title="Natural join">Natural join</a> (⨝)</td> <td>The unique relation <a href="/wiki/Relation_of_degree_zero" title="Relation of degree zero">degree zero</a> and cardinality one </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_element&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the example <i>S</i> = {<i>e,f</i>} with the equalities given, <i>S</i> is a <a href="/wiki/Semigroup" title="Semigroup">semigroup</a>. It demonstrates the possibility for <span class="texhtml">(<i>S</i>, ∗)</span> to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity. </p><p>To see this, note that if <span class="texhtml mvar" style="font-style:italic;">l</span> is a left identity and <span class="texhtml mvar" style="font-style:italic;">r</span> is a right identity, then <span class="texhtml"><i>l</i> = <i>l</i> ∗ <i>r</i> = <i>r</i></span>. In particular, there can never be more than one two-sided identity: if there were two, say <span class="texhtml mvar" style="font-style:italic;">e</span> and <span class="texhtml mvar" style="font-style:italic;">f</span>, then <span class="texhtml"><i>e</i> ∗ <i>f</i></span> would have to be equal to both <span class="texhtml mvar" style="font-style:italic;">e</span> and <span class="texhtml mvar" style="font-style:italic;">f</span>. </p><p>It is also quite possible for <span class="texhtml">(<i>S</i>, ∗)</span> to have <i>no</i> identity element,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> such as the case of even integers under the multiplication operation.<sup id="cite_ref-:0_3-2" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Another common example is the <a href="/wiki/Cross_product" title="Cross product">cross product</a> of <a href="/wiki/Euclidean_vector" title="Euclidean vector">vectors</a>, where the absence of an identity element is related to the fact that the <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">direction</a> of any nonzero cross product is always <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> to any element multiplied. That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of structure without identity element involves the additive <a href="/wiki/Semigroup" title="Semigroup">semigroup</a> of <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive</a> <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. </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=Identity_element&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Absorbing_element" title="Absorbing element">Absorbing element</a></li> <li><a href="/wiki/Additive_inverse" title="Additive inverse">Additive inverse</a></li> <li><a href="/wiki/Generalized_inverse" title="Generalized inverse">Generalized inverse</a></li> <li><a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">Identity (equation)</a></li> <li><a href="/wiki/Identity_function" title="Identity function">Identity function</a></li> <li><a href="/wiki/Inverse_element" title="Inverse element">Inverse element</a></li> <li><a href="/wiki/Monoid" title="Monoid">Monoid</a></li> <li><a href="/wiki/Pseudo-ring#Properties_weaker_than_having_an_identity" title="Pseudo-ring">Pseudo-ring</a></li> <li><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup</a></li> <li><a href="/wiki/Unital_(disambiguation)" class="mw-redirect mw-disambig" title="Unital (disambiguation)">Unital (disambiguation)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_element&action=edit&section=5" title="Edit section: Notes and references"><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 mw-references-columns"><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"><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="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/IdentityElement.html">"Identity Element"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Identity+Element&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FIdentityElement.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+element" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/identity+element">"Definition of IDENTITY ELEMENT"</a>. <i>www.merriam-webster.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.merriam-webster.com&rft.atitle=Definition+of+IDENTITY+ELEMENT&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fidentity%2Belement&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+element" class="Z3988"></span></span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/identity-element">"Identity Element"</a>. <i>www.encyclopedia.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.encyclopedia.com&rft.atitle=Identity+Element&rft_id=https%3A%2F%2Fwww.encyclopedia.com%2Fscience%2Fencyclopedias-almanacs-transcripts-and-maps%2Fidentity-element&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+element" 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"><a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 21)</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"><a href="#CITEREFBeauregardFraleigh1973">Beauregard & Fraleigh (1973</a>, p. 96)</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 18)</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFHerstein1964">Herstein (1964</a>, p. 26)</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFMcCoy1973">McCoy (1973</a>, p. 17)</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://brilliant.org/wiki/identity-element/">"Identity Element | Brilliant Math & Science Wiki"</a>. <i>brilliant.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=brilliant.org&rft.atitle=Identity+Element+%7C+Brilliant+Math+%26+Science+Wiki&rft_id=https%3A%2F%2Fbrilliant.org%2Fwiki%2Fidentity-element%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+element" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFBeauregardFraleigh1973">Beauregard & Fraleigh (1973</a>, p. 135)</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 198)</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFMcCoy1973">McCoy (1973</a>, p. 22)</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, pp. 198, 266)</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFHerstein1964">Herstein (1964</a>, p. 106)</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFMcCoy1973">McCoy (1973</a>, p. 22)</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_element&action=edit&section=6" title="Edit section: Bibliography"><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 id="CITEREFBeauregardFraleigh1973" class="citation cs2">Beauregard, Raymond A.; Fraleigh, John B. (1973), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/firstcourseinlin0000beau"><i>A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields</i></a></span>, Boston: <a href="/wiki/Houghton_Mifflin_Company" class="mw-redirect" title="Houghton Mifflin Company">Houghton Mifflin Company</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-395-14017-X" title="Special:BookSources/0-395-14017-X"><bdi>0-395-14017-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+In+Linear+Algebra%3A+with+Optional+Introduction+to+Groups%2C+Rings%2C+and+Fields&rft.place=Boston&rft.pub=Houghton+Mifflin+Company&rft.date=1973&rft.isbn=0-395-14017-X&rft.aulast=Beauregard&rft.aufirst=Raymond+A.&rft.au=Fraleigh%2C+John+B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffirstcourseinlin0000beau&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+element" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFraleigh1976" class="citation cs2">Fraleigh, John B. (1976), <i>A First Course In Abstract Algebra</i> (2nd ed.), Reading: <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-01984-1" title="Special:BookSources/0-201-01984-1"><bdi>0-201-01984-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+In+Abstract+Algebra&rft.place=Reading&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1976&rft.isbn=0-201-01984-1&rft.aulast=Fraleigh&rft.aufirst=John+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+element" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerstein1964" class="citation cs2"><a href="/wiki/I._N._Herstein" class="mw-redirect" title="I. N. Herstein">Herstein, I. N.</a> (1964), <i>Topics In Algebra</i>, Waltham: <a href="/w/index.php?title=Blaisdell_Publishing_Company&action=edit&redlink=1" class="new" title="Blaisdell Publishing Company (page does not exist)">Blaisdell Publishing Company</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1114541016" title="Special:BookSources/978-1114541016"><bdi>978-1114541016</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+In+Algebra&rft.place=Waltham&rft.pub=Blaisdell+Publishing+Company&rft.date=1964&rft.isbn=978-1114541016&rft.aulast=Herstein&rft.aufirst=I.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+element" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcCoy1973" class="citation cs2">McCoy, Neal H. (1973), <i>Introduction To Modern Algebra, Revised Edition</i>, Boston: <a href="/wiki/Allyn_and_Bacon" class="mw-redirect" title="Allyn and Bacon">Allyn and Bacon</a>, <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/68015225">68015225</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+To+Modern+Algebra%2C+Revised+Edition&rft.place=Boston&rft.pub=Allyn+and+Bacon&rft.date=1973&rft_id=info%3Alccn%2F68015225&rft.aulast=McCoy&rft.aufirst=Neal+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+element" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_element&action=edit&section=7" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>M. Kilp, U. Knauer, A.V. Mikhalev, <i>Monoids, Acts and Categories with Applications to Wreath Products and Graphs</i>, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, <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/3-11-015248-7" title="Special:BookSources/3-11-015248-7">3-11-015248-7</a>, p. 14–15</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Identity_element&oldid=1259697645">https://en.wikipedia.org/w/index.php?title=Identity_element&oldid=1259697645</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_properties_of_elements" title="Category:Algebraic properties of elements">Algebraic properties of elements</a></li><li><a href="/wiki/Category:Binary_operations" title="Category:Binary operations">Binary operations</a></li><li><a href="/wiki/Category:Properties_of_binary_operations" title="Category:Properties of binary operations">Properties of binary operations</a></li><li><a href="/wiki/Category:1_(number)" title="Category:1 (number)">1 (number)</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></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 26 November 2024, at 15:58<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. 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