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cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Edit</span> </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>additive identity</b> of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> that is equipped with the <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> of <a href="/wiki/Addition" title="Addition">addition</a> is an <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">element</a> which, when added to any element <span class="texhtml mvar" style="font-style:italic;">x</span> in the set, yields <span class="texhtml mvar" style="font-style:italic;">x</span>. One of the most familiar additive identities is the number <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0</a> from <a href="/wiki/Elementary_mathematics" title="Elementary mathematics">elementary mathematics</a>, but additive identities occur in other mathematical structures where addition is defined, such as in <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> and <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Elementary_examples"><span class="tocnumber">1</span> <span class="toctext">Elementary examples</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Formal_definition"><span class="tocnumber">2</span> <span class="toctext">Formal definition</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Further_examples"><span class="tocnumber">3</span> <span class="toctext">Further examples</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Properties"><span class="tocnumber">4</span> <span class="toctext">Properties</span></a> <ul> <li class="toclevel-2 tocsection-5"><a href="#The_additive_identity_is_unique_in_a_group"><span class="tocnumber">4.1</span> <span class="toctext">The additive identity is unique in a group</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#The_additive_identity_annihilates_ring_elements"><span class="tocnumber">4.2</span> <span class="toctext">The additive identity annihilates ring elements</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#The_additive_and_multiplicative_identities_are_different_in_a_non-trivial_ring"><span class="tocnumber">4.3</span> <span class="toctext">The additive and multiplicative identities are different in a non-trivial ring</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#See_also"><span class="tocnumber">5</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#References"><span class="tocnumber">6</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Bibliography"><span class="tocnumber">7</span> <span class="toctext">Bibliography</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#External_links"><span class="tocnumber">8</span> <span class="toctext">External links</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Elementary_examples">Elementary examples</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=1" title="Edit section: Elementary examples" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <ul><li>The additive identity familiar from <a href="/wiki/Elementary_mathematics" title="Elementary mathematics">elementary mathematics</a> is zero, denoted <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0</a>. For example, <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5+0=5=0+5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>5</mn> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5+0=5=0+5.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a1282862a8d54ebe8865918542961bfb242d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.337ex; height:2.343ex;" alt="{\displaystyle 5+0=5=0+5.}"></noscript><span class="lazy-image-placeholder" style="width: 18.337ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a1282862a8d54ebe8865918542961bfb242d53" data-alt="{\displaystyle 5+0=5=0+5.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl></li> <li>In the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" data-alt="{\displaystyle \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span> (if 0 is included), the <a href="/wiki/Integer" title="Integer">integers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.197ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" data-alt="{\displaystyle \mathbb {Z} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span> the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91185244fbdded6ea99a5e9e6603299128b10928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.455ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91185244fbdded6ea99a5e9e6603299128b10928" data-alt="{\displaystyle \mathbb {Q} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span> the <a href="/wiki/Real_number" title="Real number">real numbers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" data-alt="{\displaystyle \mathbb {R} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span> and the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" data-alt="{\displaystyle \mathbb {C} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span> the additive identity is 0. This says that for a <a href="/wiki/Number" title="Number">number</a> <span class="texhtml mvar" style="font-style:italic;">n</span> belonging to any of these sets, <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+0=n=0+n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+0=n=0+n.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb29cd5292322df7bdbbaf71b8f474a09152785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.034ex; height:2.343ex;" alt="{\displaystyle n+0=n=0+n.}"></noscript><span class="lazy-image-placeholder" style="width: 19.034ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb29cd5292322df7bdbbaf71b8f474a09152785" data-alt="{\displaystyle n+0=n=0+n.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=2" title="Edit section: Formal definition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Let <span class="texhtml mvar" style="font-style:italic;">N</span> be a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> that is closed under the <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> of <a href="/wiki/Addition" title="Addition">addition</a>, denoted <a href="/wiki/%2B" class="mw-redirect" title="+">+</a>. An additive identity for <span class="texhtml mvar" style="font-style:italic;">N</span>, denoted <span class="texhtml mvar" style="font-style:italic;">e</span>, is an element in <span class="texhtml mvar" style="font-style:italic;">N</span> such that for any element <span class="texhtml mvar" style="font-style:italic;">n</span> in <span class="texhtml mvar" style="font-style:italic;">N</span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e+n=n=n+e.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>+</mo> <mi>n</mi> <mo>=</mo> <mi>n</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mi>e</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e+n=n=n+e.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a63a45093368898370645f36c235cb52fce9553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.876ex; height:2.176ex;" alt="{\displaystyle e+n=n=n+e.}"></noscript><span class="lazy-image-placeholder" style="width: 18.876ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a63a45093368898370645f36c235cb52fce9553" data-alt="{\displaystyle e+n=n=n+e.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Further_examples">Further examples</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=3" title="Edit section: Further examples" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <ul><li>In a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, the additive identity is the <a href="/wiki/Identity_element" title="Identity element">identity element</a> of the group, is often denoted 0, and is unique (see below for proof).</li> <li>A <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> or <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the <a href="/wiki/Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a> <a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a> if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is <a href="/wiki/Trivial_(mathematics)" class="mw-redirect" title="Trivial (mathematics)">trivial</a> (proved below).</li> <li>In the ring <span class="texhtml">M<sub><i>m</i> × <i>n</i></sub>(<i>R</i>)</span> of <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> over a ring <span class="texhtml mvar" style="font-style:italic;">R</span>, the additive identity is the zero matrix,<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> denoted <span class="texhtml"><b>O</b></span> or <span class="texhtml"><b>0</b></span>, and is the <span class="texhtml mvar" style="font-style:italic;">m</span>-by-<span class="texhtml mvar" style="font-style:italic;">n</span> matrix whose entries consist entirely of the identity element 0 in <span class="texhtml mvar" style="font-style:italic;">R</span>. For example, in the 2×2 matrices over the integers <span class="nowrap">⁠<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 \operatorname {M} _{2}(\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {M} _{2}(\mathbb {Z} )}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4621ed09563a9e7c0bfe20152865fd97183d1f91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.545ex; height:2.843ex;" alt="{\displaystyle \operatorname {M} _{2}(\mathbb {Z} )}"></noscript><span class="lazy-image-placeholder" style="width: 6.545ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4621ed09563a9e7c0bfe20152865fd97183d1f91" data-alt="{\displaystyle \operatorname {M} _{2}(\mathbb {Z} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span> the additive identity is <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cab4359bab8a3b9c2b65bde9e0a3e47c7672752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.115ex; height:6.176ex;" alt="{\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 12.115ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cab4359bab8a3b9c2b65bde9e0a3e47c7672752" data-alt="{\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl></li> <li>In the <a href="/wiki/Quaternions" class="mw-redirect" title="Quaternions">quaternions</a>, 0 is the additive identity.</li> <li>In the ring of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> from <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fadf3927293c3bb66c6bb34668923799af3484b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.97ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \to \mathbb {R} }"></noscript><span class="lazy-image-placeholder" style="width: 6.97ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fadf3927293c3bb66c6bb34668923799af3484b7" data-alt="{\displaystyle \mathbb {R} \to \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span>, the function <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a> every number to 0 is the additive identity.</li> <li>In the <a href="/wiki/Additive_group" title="Additive group">additive group</a> of <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vectors</a> in <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"></noscript><span class="lazy-image-placeholder" style="width: 3.543ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" data-alt="{\displaystyle \mathbb {R} ^{n},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>⁠</span> the origin or <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a> is the additive identity.</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Properties">Properties</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=4" title="Edit section: Properties" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <div class="mw-heading mw-heading3"><h3 id="The_additive_identity_is_unique_in_a_group">The additive identity is unique in a group</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=5" title="Edit section: The additive identity is unique in a group" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Let <span class="texhtml">(<i>G</i>, +)</span> be a group and let <span class="texhtml">0</span> and <span class="texhtml">0'</span> in <span class="texhtml mvar" style="font-style:italic;">G</span> both denote additive identities, so for any <span class="texhtml mvar" style="font-style:italic;">g</span> in <span class="texhtml mvar" style="font-style:italic;">G</span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+g=g=g+0,\qquad 0'+g=g=g+0'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mi>g</mi> <mo>=</mo> <mi>g</mi> <mo>=</mo> <mi>g</mi> <mo>+</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em"></mspace> <msup> <mn>0</mn> <mo>′</mo> </msup> <mo>+</mo> <mi>g</mi> <mo>=</mo> <mi>g</mi> <mo>=</mo> <mi>g</mi> <mo>+</mo> <msup> <mn>0</mn> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+g=g=g+0,\qquad 0'+g=g=g+0'.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be50df92713f9ce38dddf28ab1cda603f0984c4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:42.797ex; height:2.843ex;" alt="{\displaystyle 0+g=g=g+0,\qquad 0'+g=g=g+0'.}"></noscript><span class="lazy-image-placeholder" style="width: 42.797ex;height: 2.843ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be50df92713f9ce38dddf28ab1cda603f0984c4f" data-alt="{\displaystyle 0+g=g=g+0,\qquad 0'+g=g=g+0'.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>It then follows from the above that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {green}0'}={\color {green}0'}+0=0'+{\color {red}0}={\color {red}0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#008000"> <msup> <mn>0</mn> <mo>′</mo> </msup> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#008000"> <msup> <mn>0</mn> <mo>′</mo> </msup> </mstyle> </mrow> <mo>+</mo> <mn>0</mn> <mo>=</mo> <msup> <mn>0</mn> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>0</mn> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>0</mn> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {green}0'}={\color {green}0'}+0=0'+{\color {red}0}={\color {red}0}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db0de04cefd279c2b544f9e5e6e8ae02c2d57b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.652ex; height:2.676ex;" alt="{\displaystyle {\color {green}0'}={\color {green}0'}+0=0'+{\color {red}0}={\color {red}0}.}"></noscript><span class="lazy-image-placeholder" style="width: 24.652ex;height: 2.676ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db0de04cefd279c2b544f9e5e6e8ae02c2d57b16" data-alt="{\displaystyle {\color {green}0'}={\color {green}0'}+0=0'+{\color {red}0}={\color {red}0}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="The_additive_identity_annihilates_ring_elements">The additive identity annihilates ring elements</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=6" title="Edit section: The additive identity annihilates ring elements" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>In a system with a multiplication operation that <a href="/wiki/Distributive_property" title="Distributive property">distributes</a> over addition, the additive identity is a multiplicative <a href="/wiki/Absorbing_element" title="Absorbing element">absorbing element</a>, meaning that for any <span class="texhtml mvar" style="font-style:italic;">s</span> in <span class="texhtml mvar" style="font-style:italic;">S</span>, <span class="texhtml"><i>s</i> · 0 = 0</span>. This follows because: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s\cdot 0&=s\cdot (0+0)=s\cdot 0+s\cdot 0\\\Rightarrow s\cdot 0&=s\cdot 0-s\cdot 0\\\Rightarrow s\cdot 0&=0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>s</mi> <mo>⋅</mo> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>s</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mi>s</mi> <mo>⋅</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> <mi>s</mi> <mo>⋅</mo> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>s</mi> <mo>⋅</mo> <mn>0</mn> <mo>−</mo> <mi>s</mi> <mo>⋅</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> <mi>s</mi> <mo>⋅</mo> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s\cdot 0&=s\cdot (0+0)=s\cdot 0+s\cdot 0\\\Rightarrow s\cdot 0&=s\cdot 0-s\cdot 0\\\Rightarrow s\cdot 0&=0.\end{aligned}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c75c0d22b1e4b64851471767535ffda955a606f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:34.298ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}s\cdot 0&=s\cdot (0+0)=s\cdot 0+s\cdot 0\\\Rightarrow s\cdot 0&=s\cdot 0-s\cdot 0\\\Rightarrow s\cdot 0&=0.\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 34.298ex;height: 8.843ex;vertical-align: -3.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c75c0d22b1e4b64851471767535ffda955a606f" data-alt="{\displaystyle {\begin{aligned}s\cdot 0&=s\cdot (0+0)=s\cdot 0+s\cdot 0\\\Rightarrow s\cdot 0&=s\cdot 0-s\cdot 0\\\Rightarrow s\cdot 0&=0.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="The_additive_and_multiplicative_identities_are_different_in_a_non-trivial_ring">The additive and multiplicative identities are different in a non-trivial ring</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=7" title="Edit section: The additive and multiplicative identities are different in a non-trivial ring" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Let <span class="texhtml mvar" style="font-style:italic;">R</span> be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let <span class="texhtml mvar" style="font-style:italic;">r</span> be any element of <span class="texhtml mvar" style="font-style:italic;">R</span>. Then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r\times 1=r\times 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>r</mi> <mo>×</mo> <mn>1</mn> <mo>=</mo> <mi>r</mi> <mo>×</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r\times 1=r\times 0=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4af23d1dfaeed7aa0b81eed4cc19b25dac9c8655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.609ex; height:2.176ex;" alt="{\displaystyle r=r\times 1=r\times 0=0}"></noscript><span class="lazy-image-placeholder" style="width: 21.609ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4af23d1dfaeed7aa0b81eed4cc19b25dac9c8655" data-alt="{\displaystyle r=r\times 1=r\times 0=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>proving that <span class="texhtml mvar" style="font-style:italic;">R</span> is trivial, i.e. <span class="texhtml"><i>R</i> = {0}.</span> The <a href="/wiki/Contrapositive" class="mw-redirect" title="Contrapositive">contrapositive</a>, that if <span class="texhtml mvar" style="font-style:italic;">R</span> is non-trivial then 0 is not equal to 1, is therefore shown. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=8" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <ul><li><a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0 (number)</a></li> <li><a href="/wiki/Additive_inverse" title="Additive inverse">Additive inverse</a></li> <li><a href="/wiki/Identity_element" title="Identity element">Identity element</a></li> <li><a href="/wiki/Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">Multiplicative identity</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=9" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AdditiveIdentity.html">"Additive Identity"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-07</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=Additive+Identity&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FAdditiveIdentity.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdditive+identity" class="Z3988"></span></span> </li> </ol></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=10" title="Edit section: Bibliography" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <ul><li>David S. Dummit, Richard M. Foote, <i>Abstract Algebra</i>, Wiley (3rd ed.): 2003, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-43334-9" title="Special:BookSources/0-471-43334-9">0-471-43334-9</a>.</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="External_links">External links</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Additive_identity&action=edit&section=11" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/UniquenessOfAdditiveIdentityInARing2">Uniqueness of additive identity in a ring</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li></ul>    </section></div> <noscript><img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" 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=Additive_identity&oldid=1253774042">https://en.wikipedia.org/w/index.php?title=Additive_identity&oldid=1253774042</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=Additive_identity&action=history"> <div class="post-content last-modified-bar__content"> <span 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अवयव – Hindi" lang="hi" hreflang="hi" data-title="योग का तत्समक अवयव" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Identitas_penambahan" title="Identitas penambahan – Indonesian" lang="id" hreflang="id" data-title="Identitas penambahan" 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-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%91%D7%A8_%D7%94%D7%90%D7%A4%D7%A1" title="איבר האפס – Hebrew" lang="he" hreflang="he" data-title="איבר האפס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Additieve_identiteit" title="Additieve identiteit – Dutch" lang="nl" hreflang="nl" data-title="Additieve identiteit" 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%8A%A0%E6%B3%95%E5%8D%98%E4%BD%8D%E5%85%83" title="加法単位元 – Japanese" lang="ja" hreflang="ja" data-title="加法単位元" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Identidade_aditiva" title="Identidade aditiva – Portuguese" lang="pt" hreflang="pt" data-title="Identidade aditiva" 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_fa%C8%9B%C4%83_de_adunare" title="Element neutru față de adunare – Romanian" lang="ro" hreflang="ro" data-title="Element neutru față de adunare" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Additive_identity" title="Additive identity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Additive identity" 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-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%82%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%B2%E0%AF%8D_%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" title="கூட்டல் முற்றொருமை – Tamil" lang="ta" hreflang="ta" data-title="கூட்டல் முற்றொருமை" 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%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%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9A%E0%B8%A7%E0%B8%81" title="เอกลักษณ์การบวก – Thai" lang="th" hreflang="th" data-title="เอกลักษณ์การบวก" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%A1n_v%E1%BB%8B_c%E1%BB%99ng" title="Đơn vị cộng – Vietnamese" lang="vi" hreflang="vi" data-title="Đơn vị cộng" 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-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8A%A0%E6%B3%95%E5%96%AE%E4%BD%8D" title="加法單位 – Cantonese" lang="yue" hreflang="yue" data-title="加法單位" 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%8A%A0%E6%B3%95%E5%96%AE%E4%BD%8D%E5%85%83" title="加法單位元 – Chinese" lang="zh" hreflang="zh" data-title="加法單位元" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page 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