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rings</span> </div> </a> <ul id="toc-Idempotents_in_split-quaternion_rings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Types_of_ring_idempotents" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_ring_idempotents"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Types of ring idempotents</span> </div> </a> <ul id="toc-Types_of_ring_idempotents-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rings_characterized_by_idempotents" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rings_characterized_by_idempotents"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Rings characterized by idempotents</span> </div> </a> <ul id="toc-Rings_characterized_by_idempotents-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Role_in_decompositions" class="vector-toc-list-item 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id="toc-Category_of_R-modules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lattice_of_idempotents" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lattice_of_idempotents"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Lattice of idempotents</span> </div> </a> <ul id="toc-Lattice_of_idempotents-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Citations</span> </div> 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Idempotent_element_(ring_theory)&redirect=no" class="mw-redirect" title="Idempotent element (ring theory)">Idempotent element (ring theory)</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">In mathematics, element that equals its square</div> <p>In <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a>, a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>idempotent element</b> or simply <b>idempotent</b> of a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> is an element <span class="texhtml"><i>a</i></span> such that <span class="texhtml"><i>a</i><sup>2</sup> = <i>a</i></span>.<sup id="cite_ref-FOOTNOTEHazewinkelGubareniKirichenko20042_1-0" class="reference"><a href="#cite_note-FOOTNOTEHazewinkelGubareniKirichenko20042-1"><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>a<span class="cite-bracket">]</span></a></sup> That is, the element is <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a> under the ring's multiplication. <a href="/wiki/Mathematical_induction" title="Mathematical induction">Inductively</a> then, one can also conclude that <span class="texhtml"><i>a</i> = <i>a</i><sup>2</sup> = <i>a</i><sup>3</sup> = <i>a</i><sup>4</sup> = ... = <i>a</i><sup><i>n</i></sup></span> for any positive <a href="/wiki/Integer" title="Integer">integer</a> <span class="texhtml"><i>n</i></span>. For example, an idempotent element of a <a href="/wiki/Matrix_ring" title="Matrix ring">matrix ring</a> is precisely an <a href="/wiki/Idempotent_matrix" title="Idempotent matrix">idempotent matrix</a>. </p><p>For general rings, elements idempotent under multiplication are involved in decompositions of <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>, and connected to <a href="/wiki/Homological_algebra" title="Homological algebra">homological</a> properties of the ring. In <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>, the main objects of study are rings in which all elements are idempotent under both addition and multiplication. </p> <meta property="mw:PageProp/toc" /> <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=Idempotent_(ring_theory)&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Quotients_of_Z">Quotients of <b>Z</b></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=2" title="Edit section: Quotients of Z"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One may consider the <a href="/wiki/Ring_of_integers_modulo_n" class="mw-redirect" title="Ring of integers modulo n">ring of integers modulo <span class="texhtml"><i>n</i></span></a>, where <span class="texhtml"><i>n</i></span> is <a href="/wiki/Square-free_integer" title="Square-free integer">squarefree</a>. By the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a>, this ring factors into the <a href="/wiki/Product_of_rings" title="Product of rings">product of rings</a> of integers modulo <span class="texhtml"><i>p</i></span>, where <span class="texhtml"><i>p</i></span> is <a href="/wiki/Prime_number" title="Prime number">prime</a>. Now each of these factors is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, so it is clear that the factors' only idempotents will be <span class="texhtml">0</span> and <span class="texhtml">1</span>. That is, each factor has two idempotents. So if there are <span class="texhtml"><i>m</i></span> factors, there will be <span class="texhtml">2<sup><i>m</i></sup></span> idempotents. </p><p>We can check this for the integers <span class="texhtml">mod 6</span>, <span class="texhtml"><i>R</i> = <b>Z</b> / 6<b>Z</b></span>. Since <span class="texhtml">6</span> has two prime factors (<span class="texhtml">2</span> and <span class="texhtml">3</span>) it should have <span class="texhtml">2<sup>2</sup></span> idempotents. </p> <dl><dd><span class="texhtml">0<sup>2</sup> ≡ 0 ≡ 0 (mod 6)</span></dd> <dd><span class="texhtml">1<sup>2</sup> ≡ 1 ≡ 1 (mod 6)</span></dd> <dd><span class="texhtml">2<sup>2</sup> ≡ 4 ≡ 4 (mod 6)</span></dd> <dd><span class="texhtml">3<sup>2</sup> ≡ 9 ≡ 3 (mod 6)</span></dd> <dd><span class="texhtml">4<sup>2</sup> ≡ 16 ≡ 4 (mod 6)</span></dd> <dd><span class="texhtml">5<sup>2</sup> ≡ 25 ≡ 1 (mod 6)</span></dd></dl> <p>From these computations, <span class="texhtml">0</span>, <span class="texhtml">1</span>, <span class="texhtml">3</span>, and <span class="texhtml">4</span> are idempotents of this ring, while <span class="texhtml">2</span> and <span class="texhtml">5</span> are not. This also demonstrates the decomposition properties described below: because <span class="texhtml">3 + 4 ≡ 1 (mod 6)</span>, there is a ring decomposition <span class="texhtml">3<b>Z</b> / 6<b>Z</b> ⊕ 4<b>Z</b> / 6<b>Z</b></span>. In <span class="texhtml">3<b>Z</b> / 6<b>Z</b></span> the multiplicative identity is <span class="texhtml">3 + 6<b>Z</b></span> and in <span class="texhtml">4<b>Z</b> / 6<b>Z</b></span> the multiplicative identity is <span class="texhtml">4 + 6<b>Z</b></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Quotient_of_polynomial_ring">Quotient of polynomial ring</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=3" title="Edit section: Quotient of polynomial ring"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a ring <span class="texhtml"><i>R</i></span> and an element <span class="texhtml"><i>f</i> ∈ <i>R</i></span> such that <span class="texhtml"><span style="padding-right:0.15em;"><i>f</i></span><sup>2</sup> ≠ 0</span>, the <a href="/wiki/Quotient_ring" title="Quotient ring">quotient ring</a> </p> <dl><dd><span class="texhtml"><i>R</i> / (<span style="padding-right:0.15em;"><i>f</i></span><sup>2</sup> − <i>f</i>)</span></dd></dl> <p>has the idempotent <span class="texhtml"><i>f</i></span>. For example, this could be applied to <span class="texhtml"><i>x</i> ∈ <b>Z</b>[<i>x</i>]</span>, or any <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> <span class="texhtml"><i>f</i> ∈ <i>k</i>[<i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub>]</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Idempotents_in_split-quaternion_rings">Idempotents in split-quaternion rings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=4" title="Edit section: Idempotents in split-quaternion rings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a <a href="/wiki/Hyperboloid" title="Hyperboloid">hyperboloid</a> of idempotents in the <a href="/wiki/Split-quaternion" title="Split-quaternion">split-quaternion</a> ring.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2023)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_ring_idempotents">Types of ring idempotents</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=5" title="Edit section: Types of ring idempotents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A partial list of important types of idempotents includes: </p> <ul><li>Two idempotents <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> are called <b>orthogonal</b> if <span class="texhtml"><i>ab</i> = <i>ba</i> = 0</span>. If <span class="texhtml"><i>a</i></span> is idempotent in the ring <span class="texhtml"><i>R</i></span> (with <a href="/wiki/Ring_(mathematics)#Notes_on_the_definition" title="Ring (mathematics)">unity</a>), then so is <span class="texhtml"><i>b</i> = 1 − <i>a</i></span>; moreover, <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> are orthogonal.</li> <li>An idempotent <span class="texhtml"><i>a</i></span> in <span class="texhtml"><i>R</i></span> is called a <b>central idempotent</b> if <span class="texhtml"><i>ax</i> = <i>xa</i></span> for all <span class="texhtml"><i>x</i></span> in <span class="texhtml"><i>R</i></span>, that is, if <span class="texhtml"><i>a</i></span> is in the <a href="/wiki/Center_(ring_theory)" title="Center (ring theory)">centre</a> of <span class="texhtml"><i>R</i></span>.</li> <li>A <b>trivial idempotent</b> refers to either of the elements <span class="texhtml">0</span> and <span class="texhtml">1</span>, which are always idempotent.</li> <li>A <b>primitive idempotent</b> of a ring <span class="texhtml"><i>R</i></span> is a nonzero idempotent <span class="texhtml"><i>a</i></span> such that <span class="texhtml"><i>aR</i></span> is <a href="/wiki/Indecomposable_module" title="Indecomposable module">indecomposable</a> as a right <span class="texhtml"><i>R</i></span>-module; that is, such that <span class="texhtml"><i>aR</i></span> is not a <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> of two <a href="/wiki/Zero_module" class="mw-redirect" title="Zero module">nonzero</a> <a href="/wiki/Submodule" class="mw-redirect" title="Submodule">submodules</a>. Equivalently, <span class="texhtml"><i>a</i></span> is a primitive idempotent if it cannot be written as <span class="texhtml"><i>a</i> = <i>e</i> + <i>f</i></span>, where <span class="texhtml"><i>e</i></span> and <span class="texhtml"><i>f</i></span> are nonzero orthogonal idempotents in <span class="texhtml"><i>R</i></span>.</li> <li>A <b>local idempotent</b> is an idempotent <span class="texhtml"><i>a</i></span> such that <span class="texhtml"><i>aRa</i></span> is a <a href="/wiki/Local_ring" title="Local ring">local ring</a>. This implies that <span class="texhtml"><i>aR</i></span> is directly indecomposable, so local idempotents are also primitive.</li> <li>A <b>right irreducible idempotent</b> is an idempotent <span class="texhtml"><i>a</i></span> for which <span class="texhtml"><i>aR</i></span> is a <a href="/wiki/Simple_module" title="Simple module">simple module</a>. By <a href="/wiki/Schur%27s_lemma" title="Schur's lemma">Schur's lemma</a>, <span class="texhtml">End<sub><i>R</i></sub>(<i>aR</i>) = <i>aRa</i></span> is a <a href="/wiki/Division_ring" title="Division ring">division ring</a>, and hence is a local ring, so right (and left) irreducible idempotents are local.</li> <li>A <b>centrally primitive</b> idempotent is a central idempotent <span class="texhtml"><i>a</i></span> that cannot be written as the sum of two nonzero orthogonal central idempotents.</li> <li>An idempotent <span class="texhtml"><i>a</i> + <i>I</i></span> in the quotient ring <span class="texhtml"><i>R</i> / <i>I</i></span> is said to <b>lift modulo <span class="texhtml"><i>I</i></span></b> if there is an idempotent <span class="texhtml"><i>b</i></span> in <span class="texhtml"><i>R</i></span> such that <span class="texhtml"><i>b</i> + <i>I</i> = <i>a</i> + <i>I</i></span>.</li> <li>An idempotent <span class="texhtml"><i>a</i></span> of <span class="texhtml"><i>R</i></span> is called a <b>full idempotent</b> if <span class="texhtml"><i>RaR</i> = <i>R</i></span>.</li> <li>A <b>separability idempotent</b>; see <i><a href="/wiki/Separable_algebra" title="Separable algebra">Separable algebra</a></i>.</li></ul> <p>Any non-trivial idempotent <span class="texhtml"><i>a</i></span> is a <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisor</a> (because <span class="texhtml"><i>ab</i> = 0</span> with neither <span class="texhtml"><i>a</i></span> nor <span class="texhtml"><i>b</i></span> being zero, where <span class="texhtml"><i>b</i> = 1 − <i>a</i></span>). This shows that <a href="/wiki/Integral_domain" title="Integral domain">integral domains</a> and <a href="/wiki/Division_ring" title="Division ring">division rings</a> do not have such idempotents. <a href="/wiki/Local_ring" title="Local ring">Local rings</a> also do not have such idempotents, but for a different reason. The only idempotent contained in the <a href="/wiki/Jacobson_radical" title="Jacobson radical">Jacobson radical</a> of a ring is <span class="texhtml">0</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Rings_characterized_by_idempotents">Rings characterized by idempotents</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=6" title="Edit section: Rings characterized by idempotents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A ring in which <i>all</i> elements are idempotent is called a <a href="/wiki/Boolean_ring" title="Boolean ring">Boolean ring</a>. Some authors use the term "idempotent ring" for this type of ring. In such a ring, multiplication is <a href="/wiki/Commutative_ring" title="Commutative ring">commutative</a> and every element is its own <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a>.</li> <li>A ring is <a href="/wiki/Semisimple_ring" class="mw-redirect" title="Semisimple ring">semisimple</a> if and only if every right (or every left) <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> is generated by an idempotent.</li> <li>A ring is <a href="/wiki/Von_Neumann_regular_ring" title="Von Neumann regular ring">von Neumann regular</a> if and only if every <a href="/wiki/Finitely_generated_module" title="Finitely generated module">finitely generated</a> right (or every finitely generated left) ideal is generated by an idempotent.</li> <li>A ring for which the <a href="/wiki/Annihilator_(ring_theory)" title="Annihilator (ring theory)">annihilator</a> <span class="texhtml"><i>r</i>.Ann(<i>S</i>)</span> every subset <span class="texhtml"><i>S</i></span> of <span class="texhtml"><i>R</i></span> is generated by an idempotent is called a <a href="/wiki/Baer_ring" title="Baer ring">Baer ring</a>. If the condition only holds for all <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">singleton</a> subsets of <span class="texhtml"><i>R</i></span>, then the ring is a right <a href="/wiki/Rickart_ring" class="mw-redirect" title="Rickart ring">Rickart ring</a>. Both of these types of rings are interesting even when they <a href="/wiki/Rng_(algebra)" title="Rng (algebra)">lack a multiplicative identity</a>.</li> <li>A ring in which all idempotents are <a href="/wiki/Center_(ring_theory)" title="Center (ring theory)">central</a> is called an <b>abelian ring</b>. Such rings need not be commutative.</li> <li>A ring is <a href="/wiki/Irreducible_ring" title="Irreducible ring">directly irreducible</a> if and only if <span class="texhtml">0</span> and <span class="texhtml">1</span> are the only central idempotents.</li> <li>A ring <span class="texhtml"><i>R</i></span> can be written as <span class="texhtml"><i>e</i><sub>1</sub><i>R</i> ⊕ <i>e</i><sub>2</sub><i>R</i> ⊕ ... ⊕ <i>e</i><sub><i>n</i></sub><i>R</i></span> with each <span class="texhtml"><i>e</i><sub><i>i</i></sub></span> a local idempotent if and only if <span class="texhtml"><i>R</i></span> is a <a href="/wiki/Semiperfect_ring" class="mw-redirect" title="Semiperfect ring">semiperfect ring</a>.</li> <li>A ring is called an <b><a href="/wiki/SBI_ring" title="SBI ring">SBI ring</a></b> or <b>Lift/rad</b> ring if all idempotents of <span class="texhtml"><i>R</i></span> lift modulo the <a href="/wiki/Jacobson_radical" title="Jacobson radical">Jacobson radical</a>.</li> <li>A ring satisfies the <a href="/wiki/Ascending_chain_condition" title="Ascending chain condition">ascending chain condition</a> on right direct summands if and only if the ring satisfies the <a href="/wiki/Descending_chain_condition" class="mw-redirect" title="Descending chain condition">descending chain condition</a> on left direct summands if and only if every set of pairwise orthogonal idempotents is finite.</li> <li>If <span class="texhtml"><i>a</i></span> is idempotent in the ring <span class="texhtml"><i>R</i></span>, then <span class="texhtml"><i>aRa</i></span> is again a ring, with multiplicative identity <span class="texhtml"><i>a</i></span>. The ring <span class="texhtml"><i>aRa</i></span> is often referred to as a <b>corner ring</b> of <span class="texhtml"><i>R</i></span>. The corner ring arises naturally since the <a href="/wiki/Ring_of_endomorphisms" class="mw-redirect" title="Ring of endomorphisms">ring of endomorphisms</a> <span class="texhtml">End<sub><i>R</i></sub>(<i>aR</i>) ≅ <i>aRa</i></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Role_in_decompositions">Role in decompositions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=7" title="Edit section: Role in decompositions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The idempotents of <span class="texhtml"><i>R</i></span> have an important connection to decomposition of <span class="texhtml"><i>R</i></span>-<a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>. If <span class="texhtml"><i>M</i></span> is an <span class="texhtml"><i>R</i></span>-module and <span class="texhtml"><i>E</i> = End<sub><i>R</i></sub>(<i>M</i>)</span> is its <a href="/wiki/Ring_of_endomorphisms" class="mw-redirect" title="Ring of endomorphisms">ring of endomorphisms</a>, then <span class="texhtml"><i>A</i> ⊕ <i>B</i> = <i>M</i></span> if and only if there is a unique idempotent <span class="texhtml"><i>e</i></span> in <span class="texhtml"><i>E</i></span> such that <span class="texhtml"><i>A</i> = <i>eM</i></span> and <span class="texhtml"><i>B</i> = (1 − <i>e</i>)<i>M</i></span>. Clearly then, <span class="texhtml"><i>M</i></span> is directly indecomposable if and only if <span class="texhtml">0</span> and <span class="texhtml">1</span> are the only idempotents in <span class="texhtml"><i>E</i></span>.<sup id="cite_ref-FOOTNOTEAndersonFuller1992p._69–72_3-0" class="reference"><a href="#cite_note-FOOTNOTEAndersonFuller1992p._69–72-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In the case when <span class="texhtml"><i>M</i> = <i>R</i></span> (assumed unital), the endomorphism ring <span class="texhtml">End<sub><i>R</i></sub>(<i>R</i>) = <i>R</i></span>, where each <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> arises as left multiplication by a fixed ring element. With this modification of notation, <span class="texhtml"><i>A</i> ⊕ <i>B</i> = <i>R</i></span> as right modules if and only if there exists a unique idempotent <span class="texhtml"><i>e</i></span> such that <span class="texhtml"><i>eR</i> = <i>A</i></span> and <span class="texhtml">(1 − <i>e</i>)<i>R</i> = <i>B</i></span>. Thus every direct summand of <span class="texhtml"><i>R</i></span> is generated by an idempotent. </p><p>If <span class="texhtml"><i>a</i></span> is a central idempotent, then the corner ring <span class="texhtml"><i>aRa</i> = <i>Ra</i></span> is a ring with multiplicative identity <span class="texhtml"><i>a</i></span>. Just as idempotents determine the direct decompositions of <span class="texhtml"><i>R</i></span> as a module, the central idempotents of <span class="texhtml"><i>R</i></span> determine the decompositions of <span class="texhtml"><i>R</i></span> as a <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> of rings. If <span class="texhtml"><i>R</i></span> is the direct sum of the rings <span class="texhtml"><i>R</i><sub>1</sub></span>, ..., <span class="texhtml"><i>R</i><sub><i>n</i></sub></span>, then the identity elements of the rings <span class="texhtml"><i>R</i><sub><i>i</i></sub></span> are central idempotents in <span class="texhtml"><i>R</i></span>, pairwise orthogonal, and their sum is <span class="texhtml">1</span>. Conversely, given central idempotents <span class="texhtml"><i>a</i><sub>1</sub></span>, ..., <span class="texhtml"><i>a</i><sub><i>n</i></sub></span> in <span class="texhtml"><i>R</i></span> that are pairwise orthogonal and have sum <span class="texhtml">1</span>, then <span class="texhtml"><i>R</i></span> is the direct sum of the rings <span class="texhtml"><i>Ra</i><sub>1</sub></span>, ..., <span class="texhtml"><i>Ra</i><sub><i>n</i></sub></span>. So in particular, every central idempotent <span class="texhtml"><i>a</i></span> in <span class="texhtml"><i>R</i></span> gives rise to a decomposition of <span class="texhtml"><i>R</i></span> as a direct sum of the corner rings <span class="texhtml"><i>aRa</i></span> and <span class="texhtml">(1 − <i>a</i>)<i>R</i>(1 − <i>a</i>)</span>. As a result, a ring <span class="texhtml"><i>R</i></span> is directly indecomposable as a ring if and only if the identity <span class="texhtml">1</span> is centrally primitive. </p><p>Working inductively, one can attempt to decompose <span class="texhtml">1</span> into a sum of centrally primitive elements. If <span class="texhtml">1</span> is centrally primitive, we are done. If not, it is a sum of central orthogonal idempotents, which in turn are primitive or sums of more central idempotents, and so on. The problem that may occur is that this may continue without end, producing an infinite family of central orthogonal idempotents. The condition "<span class="texhtml"><i>R</i></span><i> does not contain infinite sets of central orthogonal idempotents</i>" is a type of finiteness condition on the ring. It can be achieved in many ways, such as requiring the ring to be right <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian</a>. If a decomposition <span class="texhtml"><i>R</i> = <i>c</i><sub>1</sub><i>R</i> ⊕ <i>c</i><sub>2</sub><i>R</i> ⊕ ... ⊕ <i>c</i><sub><i>n</i></sub><i>R</i></span> exists with each <span class="texhtml"><i>c</i><sub><i>i</i></sub></span> a centrally primitive idempotent, then <span class="texhtml"><i>R</i></span> is a direct sum of the corner rings <span class="texhtml"><i>c</i><sub><i>i</i></sub><i>Rc</i><sub><i>i</i></sub></span>, each of which is ring irreducible.<sup id="cite_ref-FOOTNOTELam2001p._326_4-0" class="reference"><a href="#cite_note-FOOTNOTELam2001p._326-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>For <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebras</a> or <a href="/wiki/Jordan_algebra" title="Jordan algebra">Jordan algebras</a> over a field, the <a href="/wiki/Peirce_decomposition" title="Peirce decomposition">Peirce decomposition</a> is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_with_involutions">Relation with involutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=8" title="Edit section: Relation with involutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml"><i>a</i></span> is an idempotent of the endomorphism ring <span class="texhtml">End<sub><i>R</i></sub>(<i>M</i>)</span>, then the endomorphism <span class="texhtml"><i>f</i> = 1 − 2<i>a</i></span> is an <span class="texhtml"><i>R</i></span>-module <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> of <span class="texhtml"><i>M</i></span>. That is, <span class="texhtml"><i>f</i></span> is an <span class="texhtml"><i>R</i></span>-<a href="/wiki/Module_homomorphism" title="Module homomorphism">module homomorphism</a> such that <span class="texhtml"><span style="padding-right:0.15em;"><i>f</i></span><sup>2</sup></span> is the identity endomorphism of <span class="texhtml"><i>M</i></span>. </p><p>An idempotent element <span class="texhtml"><i>a</i></span> of <span class="texhtml"><i>R</i></span> and its associated involution <span class="texhtml"><i>f</i></span> gives rise to two involutions of the module <span class="texhtml"><i>R</i></span>, depending on viewing <span class="texhtml"><i>R</i></span> as a left or right module. If <span class="texhtml"><i>r</i></span> represents an arbitrary element of <span class="texhtml"><i>R</i></span>, <span class="texhtml"><i>f</i></span> can be viewed as a right <span class="texhtml"><i>R</i></span>-module homomorphism <span class="texhtml"><i>r</i> ↦ <i>fr</i></span> so that <span class="texhtml"><i>ffr</i> = <i>r</i></span>, or <span class="texhtml"><i>f</i></span> can also be viewed as a left <span class="texhtml"><i>R</i></span>-module homomorphism <span class="texhtml"><i>r</i> ↦ <i>rf</i></span>, where <span class="texhtml"><i>rff</i> = <i>r</i></span>. </p><p>This process can be reversed if <span class="texhtml">2</span> is an <a href="/wiki/Invertible_element" class="mw-redirect" title="Invertible element">invertible element</a> of <span class="texhtml"><i>R</i></span>:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> if <span class="texhtml"><i>b</i></span> is an involution, then <span class="texhtml">2<sup>−1</sup>(1 − <i>b</i>)</span> and <span class="texhtml">2<sup>−1</sup>(1 + <i>b</i>)</span> are orthogonal idempotents, corresponding to <span class="texhtml"><i>a</i></span> and <span class="texhtml">1 − <i>a</i></span>. Thus for a ring in which <span class="texhtml">2</span> is invertible, the idempotent elements <a href="/wiki/Bijection" title="Bijection">correspond</a> to involutions in a one-to-one manner. </p> <div class="mw-heading mw-heading2"><h2 id="Category_of_R-modules">Category of <i>R</i>-modules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=9" title="Edit section: Category of R-modules"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lifting idempotents also has major consequences for the <a href="/wiki/Category_of_modules" title="Category of modules">category of <span class="texhtml"><i>R</i></span>-modules</a>. All idempotents lift modulo <span class="texhtml"><i>I</i></span> if and only if every <span class="texhtml"><i>R</i></span> direct summand of <span class="texhtml"><i>R</i>/<i>I</i></span> has a <a href="/wiki/Projective_cover" title="Projective cover">projective cover</a> as an <span class="texhtml"><i>R</i></span>-module.<sup id="cite_ref-FOOTNOTEAndersonFuller1992p._302_6-0" class="reference"><a href="#cite_note-FOOTNOTEAndersonFuller1992p._302-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Idempotents always lift modulo <a href="/wiki/Nil_ideal" title="Nil ideal">nil ideals</a> and rings for which <span class="texhtml"><i>R</i></span> is <a href="/wiki/Completion_(ring_theory)#Krull_topology" class="mw-redirect" title="Completion (ring theory)"><span class="texhtml"><i>I</i></span>-adically complete</a>. </p><p>Lifting is most important when <span class="texhtml"><i>I</i> = J(<i>R</i>)</span>, the <a href="/wiki/Jacobson_radical" title="Jacobson radical">Jacobson radical</a> of <span class="texhtml"><i>R</i></span>. Yet another characterization of <a href="/wiki/Semiperfect_ring" class="mw-redirect" title="Semiperfect ring">semiperfect rings</a> is that they are <a href="/wiki/Semilocal_ring" class="mw-redirect" title="Semilocal ring">semilocal rings</a> whose idempotents lift modulo <span class="texhtml">J(<i>R</i>)</span>.<sup id="cite_ref-FOOTNOTELam2001p._336_7-0" class="reference"><a href="#cite_note-FOOTNOTELam2001p._336-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Lattice_of_idempotents">Lattice of idempotents</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=10" title="Edit section: Lattice of idempotents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One may define a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> on the idempotents of a ring as follows: if <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> are idempotents, we write <span class="texhtml"><i>a</i> ≤ <i>b</i></span> if and only if <span class="texhtml"><i>ab</i> = <i>ba</i> = <i>a</i></span>. With respect to this order, <span class="texhtml">0</span> is the smallest and <span class="texhtml">1</span> the largest idempotent. For orthogonal idempotents <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span>, <span class="texhtml"><i>a</i> + <i>b</i></span> is also idempotent, and we have <span class="texhtml"><i>a</i> ≤ <i>a</i> + <i>b</i></span> and <span class="texhtml"><i>b</i> ≤ <i>a</i> + <i>b</i></span>. The <a href="/wiki/Atom_(order_theory)" title="Atom (order theory)">atoms</a> of this partial order are precisely the primitive idempotents.<sup id="cite_ref-FOOTNOTELam2001323_8-0" class="reference"><a href="#cite_note-FOOTNOTELam2001323-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>When the above partial order is restricted to the central idempotents of <span class="texhtml"><i>R</i></span>, a <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> structure, or even a <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a> structure, can be given. For two central idempotents <span class="texhtml"><i>e</i></span> and <span class="texhtml"><i>f</i></span>, the <a href="/wiki/Boolean_algebra#Operations" title="Boolean algebra">complement</a> is given by </p> <dl><dd><span class="texhtml">¬<i>e</i> = 1 − <i>e</i></span>,</dd></dl> <p>the <a href="/wiki/Meet_(mathematics)" class="mw-redirect" title="Meet (mathematics)">meet</a> is given by </p> <dl><dd><span class="texhtml"><i>e</i> ∧ <i>f</i> = <i>ef</i></span>.</dd></dl> <p>and the <a href="/wiki/Join_(mathematics)" class="mw-redirect" title="Join (mathematics)">join</a> is given by </p> <dl><dd><span class="texhtml"><i>e</i> ∨ <i>f</i> = ¬(¬<i>e</i> ∧ ¬<i>f</i>) = <i>e</i> + <i>f</i> − <i>ef</i></span></dd></dl> <p>The ordering now becomes simply <span class="texhtml"><i>e</i> ≤ <i>f</i></span> if and only if <span class="texhtml"><i>eR</i> ⊆ <span style="padding-right:0.15em;"><i>f</i></span><i>R</i></span>, and the join and meet satisfy <span class="texhtml">(<i>e</i> ∨ <span style="padding-right:0.15em;"><i>f</i></span>)<i>R</i> = <i>eR</i> + <span style="padding-right:0.15em;"><i>f</i></span><i>R</i></span> and <span class="texhtml">(<i>e</i> ∧ <span style="padding-right:0.15em;"><i>f</i></span>)<i>R</i> = <i>eR</i> ∩ <span style="padding-right:0.15em;"><i>f</i></span><i>R</i> = (<i>eR</i>)(<span style="padding-right:0.15em;"><i>f</i></span><i>R</i>)</span>. It is shown in <a href="#CITEREFGoodearl1991">Goodearl 1991</a>, p. 99 that if <span class="texhtml"><i>R</i></span> is <a href="/wiki/Von_Neumann_regular" class="mw-redirect" title="Von Neumann regular">von Neumann regular</a> and right <a href="/wiki/Injective_module#Self-injective_rings" title="Injective module">self-injective</a>, then the lattice is a <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattice</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Idempotent and <a href="/wiki/Nilpotent" title="Nilpotent">nilpotent</a> were introduced by <a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a> in 1870.</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">Rings in which <span class="texhtml">2</span> is not invertible are not difficult to find. The element <span class="texhtml">2</span> is not invertible in any ring of <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> <span class="texhtml">2</span>, which includes <a href="/wiki/Boolean_ring" title="Boolean ring">Boolean rings</a>.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The restriction "invertible" should presumably be replaced by "cancellable": an example where <span class="texhtml " >2</span> is not invertible, but this process works through division is in the ring of integers. (October 2023)">clarification needed</span></a></i>]</sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=12" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-FOOTNOTEHazewinkelGubareniKirichenko20042-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHazewinkelGubareniKirichenko20042_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHazewinkelGubareniKirichenko2004">Hazewinkel, Gubareni & Kirichenko 2004</a>, p. 2</span> </li> <li id="cite_note-FOOTNOTEAndersonFuller1992p._69–72-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAndersonFuller1992p._69–72_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAndersonFuller1992">Anderson & Fuller 1992</a>, p. 69–72</span> </li> <li id="cite_note-FOOTNOTELam2001p._326-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELam2001p._326_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLam2001">Lam 2001</a>, p. 326</span> </li> <li id="cite_note-FOOTNOTEAndersonFuller1992p._302-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAndersonFuller1992p._302_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAndersonFuller1992">Anderson & Fuller 1992</a>, p. 302</span> </li> <li id="cite_note-FOOTNOTELam2001p._336-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELam2001p._336_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLam2001">Lam 2001</a>, p. 336</span> </li> <li id="cite_note-FOOTNOTELam2001323-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELam2001323_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLam2001">Lam 2001</a>, p. 323</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Idempotent_(ring_theory)&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><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="CITEREFAndersonFuller1992" class="citation cs2">Anderson, Frank Wylie; Fuller, Kent R (1992), <i>Rings and Categories of Modules</i>, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97845-1" title="Special:BookSources/978-0-387-97845-1"><bdi>978-0-387-97845-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rings+and+Categories+of+Modules&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1992&rft.isbn=978-0-387-97845-1&rft.aulast=Anderson&rft.aufirst=Frank+Wylie&rft.au=Fuller%2C+Kent+R&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdempotent+%28ring+theory%29" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://foldoc.org/idempotent">idempotent</a> at <a href="/wiki/FOLDOC" class="mw-redirect" title="FOLDOC">FOLDOC</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodearl1991" class="citation cs2">Goodearl, K. R. (1991), <i>von Neumann regular rings</i> (2 ed.), Malabar, FL: Robert E. Krieger Publishing Co. Inc., pp. xviii+412, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89464-632-X" title="Special:BookSources/0-89464-632-X"><bdi>0-89464-632-X</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1150975">1150975</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=von+Neumann+regular+rings&rft.place=Malabar%2C+FL&rft.pages=xviii%2B412&rft.edition=2&rft.pub=Robert+E.+Krieger+Publishing+Co.+Inc.&rft.date=1991&rft.isbn=0-89464-632-X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1150975%23id-name%3DMR&rft.aulast=Goodearl&rft.aufirst=K.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdempotent+%28ring+theory%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHazewinkelGubareniKirichenko2004" class="citation cs2">Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2004), <i>Algebras, rings and modules. Vol. 1</i>, Mathematics and its Applications, vol. 575, Dordrecht: Kluwer Academic Publishers, pp. xii+380, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-2690-0" title="Special:BookSources/1-4020-2690-0"><bdi>1-4020-2690-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2106764">2106764</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebras%2C+rings+and+modules.+Vol.+1&rft.place=Dordrecht&rft.series=Mathematics+and+its+Applications&rft.pages=xii%2B380&rft.pub=Kluwer+Academic+Publishers&rft.date=2004&rft.isbn=1-4020-2690-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2106764%23id-name%3DMR&rft.aulast=Hazewinkel&rft.aufirst=Michiel&rft.au=Gubareni%2C+Nadiya&rft.au=Kirichenko%2C+V.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdempotent+%28ring+theory%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLam2001" class="citation cs2">Lam, T. Y. (2001), <i>A first course in noncommutative rings</i>, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4419-8616-0">10.1007/978-1-4419-8616-0</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95183-0" title="Special:BookSources/0-387-95183-0"><bdi>0-387-95183-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1838439">1838439</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+noncommutative+rings&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pages=xx%2B385&rft.edition=2&rft.pub=Springer-Verlag&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1838439%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4419-8616-0&rft.isbn=0-387-95183-0&rft.aulast=Lam&rft.aufirst=T.+Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdempotent+%28ring+theory%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1993" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1993), <i>Algebra</i> (Third ed.), Reading, Mass.: Addison-Wesley, p. 443, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-55540-0" title="Special:BookSources/978-0-201-55540-0"><bdi>978-0-201-55540-0</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0848.13001">0848.13001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.place=Reading%2C+Mass.&rft.pages=443&rft.edition=Third&rft.pub=Addison-Wesley&rft.date=1993&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0848.13001%23id-name%3DZbl&rft.isbn=978-0-201-55540-0&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdempotent+%28ring+theory%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeirce1870" class="citation cs2">Peirce, Benjamin (1870), <a rel="nofollow" class="external text" href="http://www.math.harvard.edu/history/peirce_algebra/index.html"><i>Linear Associative Algebra</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Associative+Algebra&rft.date=1870&rft.aulast=Peirce&rft.aufirst=Benjamin&rft_id=http%3A%2F%2Fwww.math.harvard.edu%2Fhistory%2Fpeirce_algebra%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdempotent+%28ring+theory%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolcino_MiliesSehgal2002" class="citation cs2">Polcino Milies, César; Sehgal, Sudarshan K. (2002), <i>An introduction to group rings</i>, Algebras and Applications, vol. 1, Dordrecht: Kluwer Academic Publishers, pp. xii+371, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-010-0405-3">10.1007/978-94-010-0405-3</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-0238-6" title="Special:BookSources/1-4020-0238-6"><bdi>1-4020-0238-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1896125">1896125</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+group+rings&rft.place=Dordrecht&rft.series=Algebras+and+Applications&rft.pages=xii%2B371&rft.pub=Kluwer+Academic+Publishers&rft.date=2002&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1896125%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-94-010-0405-3&rft.isbn=1-4020-0238-6&rft.aulast=Polcino+Milies&rft.aufirst=C%C3%A9sar&rft.au=Sehgal%2C+Sudarshan+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdempotent+%28ring+theory%29" class="Z3988"></span></li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Idempotent_(ring_theory)&oldid=1223213818">https://en.wikipedia.org/w/index.php?title=Idempotent_(ring_theory)&oldid=1223213818</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">Category</a>: <ul><li><a href="/wiki/Category:Ring_theory" title="Category:Ring theory">Ring theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_October_2023" title="Category:Wikipedia articles needing clarification from October 2023">Wikipedia articles needing clarification from October 2023</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_October_2023" title="Category:Articles with unsourced statements from October 2023">Articles with unsourced statements from October 2023</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 10 May 2024, at 16:34<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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