CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> ring in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> ring </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4173/#Item_24" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> <h4 id="linear_algebra">Linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#generalizations'>Generalizations</a></li> <ul> <li><a href='#internalising_the_sets'>Internalising the sets</a></li> <li><a href='#internalising_the_abelian_groups'>Internalising the abelian groups</a></li> <li><a href='#_rings'><a class="existingWikiWord" href="/nlab/show/ring+over+a+ring">Rings over a ring</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-rings)</a></li> <li><a href='#higher_rings'>Higher rings</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#types_of_rings'>Types of rings</a></li> <ul> <li><a href='#integral_domains'>Integral domains</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#ReferencesGeneral'>General</a></li> <li><a href='#ReferencesHistory'>History</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>A <em>ring</em> (also: <em>number ring</em>) is a basic structure in <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>: a <a class="existingWikiWord" href="/nlab/show/set">set</a> equipped with two <a class="existingWikiWord" href="/nlab/show/binary+operation"> binary operations</a> called <em>addition</em> and <em>multiplication</em>, such that the operation of addition forms an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> and the operation of multiplication a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> structure which <a class="existingWikiWord" href="/nlab/show/distributivity+law">distributes</a> over addition.</p> <p>All the familiar number systems such as the <a class="existingWikiWord" href="/nlab/show/integer+numbers">integer numbers</a>, <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>, <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, <a class="existingWikiWord" href="/nlab/show/complex">complex</a> numbers are rings under the standard operations of addition and multiplication. Except for the first in this list they are indeed <a class="existingWikiWord" href="/nlab/show/fields">fields</a>, which are rings in which also the multiplication operation has an inverse for every element except 0 (the additive neutral element).</p> <p>Other basic examples of rings are the <a class="existingWikiWord" href="/nlab/show/cyclic+groups">cyclic groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_n</annotation></semantics></math> under their mod-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> operations inherited from the integers (<a class="existingWikiWord" href="/nlab/show/integers+modulo+n">integers modulo n</a>); the <a class="existingWikiWord" href="/nlab/show/polynomial+rings">polynomial rings</a>, etc.</p> <p>More abstractly, a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> is a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> (with their <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a>), and this perspective helps to explain the central relevance of the concept, owing to the fundamental nature of the notion of <em><a class="existingWikiWord" href="/nlab/show/monoid+objects">monoid objects</a></em>. Accordingly monoids internal to other <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> and more generally <a class="existingWikiWord" href="/nlab/show/stable+infinity-categories">stable infinity-categories</a> constitute generalizations of the notion of <em>ring</em> that are of interest. Notably when abelian groups are generalized to their analogs in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>, namely to <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a>, the corresponding internal monoids are <a class="existingWikiWord" href="/nlab/show/E-infinity+rings">E-infinity rings</a>, a basic structure in <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a>.</p> <p>Rings form a category, <a class="existingWikiWord" href="/nlab/show/Ring">Ring</a>, which contains the category of commutative rings, <a class="existingWikiWord" href="/nlab/show/CRing">CRing</a>, as a subcategory.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/ring">ring</a> (unital and not-necessarily commutative) is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> equipped with</p> <ol> <li> <p>an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">1 \in R</annotation></semantics></math></p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a>, hence a <a class="existingWikiWord" href="/nlab/show/group+homomorphism">group homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⋅</mo><mo>:</mo><mi>R</mi><mo>⊗</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex"> \cdot : R \otimes R \to R </annotation></semantics></math></div> <p>out of the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a>,</p> </li> </ol> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/associativity+law">associative</a> and <a class="existingWikiWord" href="/nlab/show/unit+law">unital</a> with respect to 1.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The fact that the product is a <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a> is the <strong><a class="existingWikiWord" href="/nlab/show/distributivity+law">distributivity law</a></strong>: for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>,</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>r</mi> <mn>2</mn></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r, r_1, r_2 \in R</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mo>⋅</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>+</mo><mi>r</mi><mo>⋅</mo><msub><mi>r</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> r \cdot (r_1 + r_2) = r \cdot r_1 + r \cdot r_2 </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><mi>r</mi><mo>=</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>⋅</mo><mi>r</mi><mo>+</mo><msub><mi>r</mi> <mn>2</mn></msub><mo>⋅</mo><mi>r</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (r_1 + r_2) \cdot r = r_1 \cdot r + r_2 \cdot r \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A (unital, non-commutative) <strong>ring</strong> is (equivalently)</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> regarded as a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> equipped with the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a>;</li> <li>a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> <a class="existingWikiWord" href="/nlab/show/enriched+category">category enriched over</a> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> with a single <a class="existingWikiWord" href="/nlab/show/object">object</a>.</li> <li>a <a class="existingWikiWord" href="/nlab/show/ringoid">ringoid</a> with a single <a class="existingWikiWord" href="/nlab/show/object">object</a>.</li> </ul> <p>A <strong>commutative</strong> (unital) ring is a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ab</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Ab, \otimes)</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>In usual ring theory people often talk about <strong><a class="existingWikiWord" href="/nlab/show/nonunital+rings">nonunital rings</a></strong> as well: multiplicative <a class="existingWikiWord" href="/nlab/show/semigroups">semigroups</a> with additive <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> structure where the multiplication is distributive toward addition; these are semigroup objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>. As in the unital case, if the semigroup is abelian then the ring is said to be <strong>commutative nonunital</strong>. Note the adjective ‘nonunital’ is an example of the <a class="existingWikiWord" href="/nlab/show/red+herring+principle">red herring principle</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>If one removes the assumption that the additive group is abelian but retains the remaining ring axioms, the result is still a ring. More generally, this holds for nonunital rings of which multiplicative semigroup is left/right <a class="existingWikiWord" href="/nlab/show/weakly+reductive+semigroup">weakly reductive</a>. The result is false for arbitrary nonunital rings: for any group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G, +, 0)</annotation></semantics></math> we could define multiplication to be the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⋅</mo><mi>y</mi><mo>:</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x\cdot y := 0</annotation></semantics></math>, and all the axioms except additive commutativity are trivially satisfied. This occurs because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math> doesn’t distinguish between elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </div> <h2 id="generalizations">Generalizations</h2> <p>It is possible to <a class="existingWikiWord" href="/nlab/show/internalization">internalise</a> the notion of ring in at least two different ways. Either one can replace the <a class="existingWikiWord" href="/nlab/show/Set">category of sets</a> in the classical definition with another category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> – see <a class="existingWikiWord" href="/nlab/show/ring+object">ring object</a> – , or one can replace <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> in the fancy definition with another category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>.</p> <h3 id="internalising_the_sets">Internalising the sets</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a>, then any <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> may be internalised in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. The theory of rings is an example, so we can speak of <em>ring objects</em> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Then a ring object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> is simply a ring. (This works whether your rings are unital or nonunital, commutative or noncommutative, etc.) However, not every notion of internal ring takes this form.</p> <p>The theory of rings is a combination of a monoid (or semigroup, if nonunital) and an abelian group structure. Thus, ring objects are algebras over a composed <a class="existingWikiWord" href="/nlab/show/monad">monad</a> of a monoid monad and an abelian group monad, using a <a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a>, which corresponds to the usual distributive law in the classical definition of a ring.</p> <p>A particular example of this is a ring in a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>. In a topos one usually alternatively defines a ring object by the standard set-theoretic definition of a ring, and interpret the formulas in the sense of topos-theoretic semantics.</p> <p>Picking a ring object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> promotes it into a <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a>.</p> <p>In cartesian categories one can also define the structure of an (abelian) group object as the lifting of the correspoding <a class="existingWikiWord" href="/nlab/show/representable+presheaf">representable presheaf</a> to a presheaf into (abelian) groups. This kind of lifting of some algebraic structure in sets to algebraic structure in a cartesian category makes sense when some category of algebras creates the limits needed to define them in sets.</p> <h3 id="internalising_the_abelian_groups">Internalising the abelian groups</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, then we can speak of <a class="existingWikiWord" href="/nlab/show/monoid+objects">monoid objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. However, we usually want <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> to be somewhat like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math> to think of monoid objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> as internal rings. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is the category of abelian <a class="existingWikiWord" href="/nlab/show/group+objects">group objects</a> in a cartesian monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, then we recreate the notion of ring object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> from above. Or, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is any <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a>, then it behaves enough like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math> that we may consider its monoid objects as internal rings. There are yet other examples, however: a <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a> is a monoid object in <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a>, even though these are not <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>-enriched.</p> <p>Other examples are <a class="existingWikiWord" href="/nlab/show/simplicial+ring"> simplicial rings</a> (as monoids in <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group"> simplicial abelian groups</a>) and <a class="existingWikiWord" href="/nlab/show/dg-ring"> dg-rings</a>, as well as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-rings below.</p> <h3 id="_rings"><a class="existingWikiWord" href="/nlab/show/ring+over+a+ring">Rings over a ring</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-rings)</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a commutative ring (or especially a <a class="existingWikiWord" href="/nlab/show/field">field</a>), then an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a monoid object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Mod">Mod</a>; this is a special case of the previous section.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a noncommutative ring, then a <strong>ring over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></strong>, or simply an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-ring</strong>, is a monoid object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Bimod">Bimod</a> (that is, in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo></mo><mi>A</mi></msub><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">_A Mod _A</annotation></semantics></math>). Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-ring is a ring in the usual sense, in the sense that there is an obvious <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> to the usual rings. In fact the unit map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">A \to R</annotation></semantics></math> is a morphism of rings, and the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-rings is precisely the <a class="existingWikiWord" href="/nlab/show/coslice+category">coslice category</a> or under-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mi>Ring</mi></mrow><annotation encoding="application/x-tex">A/Ring</annotation></semantics></math>. Thus by category-theoretic rules, one might be led to unconventionally call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-rings “rings <em>under</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>”. Unfortunately, standard name for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-rings is “rings <em>over</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>”, like conventionally calling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-algebras the “algebras <em>over</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>”.</p> <p>Unlike for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-algebras, the multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R\times R\to R</annotation></semantics></math> which is the morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bimodules">bimodules</a>, is not (left) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-linear in the <em>second</em> factor, but only <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A^{op}</annotation></semantics></math>-linear (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-linear on the right). In other words, the axiom for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mo stretchy="false">(</mo><mi>k</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k (r s) = r (k s)</annotation></semantics></math> is not true, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">k\in A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>,</mo><mi>s</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r,s\in R</annotation></semantics></math>, although <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mi>r</mi><mo stretchy="false">)</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">k (r s) = (k r) s</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mi>k</mi><mo>=</mo><mi>r</mi><mo stretchy="false">(</mo><mi>s</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(r s) k = r (s k)</annotation></semantics></math> do hold.</p> <p>Both for a discussion for under-over and also for this difference between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-algebras and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-rings see the Café's <a href="http://golem.ph.utexas.edu/category/2008/12/a_quick_algebra_quiz.html">quick algebra quiz</a>.</p> <p>A dual notion to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-ring is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/coring">coring</a>.</p> <p>The structure of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A\otimes A^{op}</annotation></semantics></math>-ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(R,\mu,\eta)</annotation></semantics></math> is determined by the structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as a ring, together with the two natural homomorphisms of rings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>=</mo><mi>η</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⊗</mo><msub><mn>1</mn> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">s = \eta(-\otimes 1_A):A\to R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>=</mo><mi>η</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>A</mi></msub><mo>⊗</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">t=\eta(1_A\otimes -):A^{op}\to R</annotation></semantics></math> which have commuting images (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s(a)t(a')=t(a')s(a)</annotation></semantics></math>, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a,a'\in A</annotation></semantics></math>).</p> <h3 id="higher_rings">Higher rings</h3> <p>By replacing in the sentence “a ring is a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>” the <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> with a <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category</a> of <em>symmetric monoidal</em> higher groupoids, one obtains higher notion of rings, such as a <a class="existingWikiWord" href="/nlab/show/ring+groupoid">ring groupoid</a>, or in the commutative case, a <a class="existingWikiWord" href="/nlab/show/symmetric+ring+groupoid">symmetric ring groupoid</a>.</p> <p>Of particular interest is the maximal case of symmetric monoidal <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid"> ∞-groupoids</a> and, even more generally, that of <a class="existingWikiWord" href="/nlab/show/spectrum">spectra</a>. A <a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a> in the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a> is an <a class="existingWikiWord" href="/nlab/show/A-infinity-ring">A-infinity-ring</a> or <a class="existingWikiWord" href="/nlab/show/associative+ring+spectrum">associative ring spectrum</a>. The commutative case is a <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a>: an <a class="existingWikiWord" href="/nlab/show/E-infinity+ring">E-infinity ring</a> or <a class="existingWikiWord" href="/nlab/show/commutative+ring+spectrum">commutative ring spectrum</a>.</p> <h2 id="Examples">Examples</h2> <div class="num_example"> <h6 id="example">Example</h6> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> are a ring under the standard addition and multiplication operation.</p> </li> <li> <p>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, this induces a ring structure on the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_n</annotation></semantics></math>, given by operations in the <a class="existingWikiWord" href="/nlab/show/integers+modulo+n">integers modulo n</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> are rings under their standard operations, in fact these are even <em><a class="existingWikiWord" href="/nlab/show/fields">fields</a></em>.</p> </li> </ul> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a ring, the <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>r</mi> <mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>r</mi> <mn>2</mn></msub><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>r</mi> <mi>n</mi></msub><msup><mi>x</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> r_0 + r_1 x + r_2 x^2 + \cdots + r_n x^n </annotation></semantics></math></div> <p>(for arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in\mathbb{N}</annotation></semantics></math>) in a <a class="existingWikiWord" href="/nlab/show/variable">variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> form another ring, the <em><a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a></em> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R[x]</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> on a single generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a ring and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(n,R)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/matrices">matrices</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a ring under elementwise addition and <a class="existingWikiWord" href="/nlab/show/matrix+multiplication">matrix multiplication</a>.</p> </div> <div class="num_example"> <h6 id="example_4">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the set of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\mathbb{R})</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\mathbb{C})</annotation></semantics></math> with values in the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> or <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> is a ring under pointwise (points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>) addition and multiplication.</p> </div> <div class="num_example"> <h6 id="example_5">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of its <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet(X,\mathbb{Z})</annotation></semantics></math> forms a ring whose multiplication operation is the <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a>. This is a <a class="existingWikiWord" href="/nlab/show/graded+object">graded ring</a>, graded by the cohomological degree.</p> </div> <h2 id="types_of_rings">Types of rings</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GCD+ring">GCD ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B%C3%A9zout+ring">Bézout ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+ring">Euclidean ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prefield+ring">prefield ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/zero-dimensional+ring">zero-dimensional ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+ring">principal ideal ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+ring">quotient ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/composition+ring">composition ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+ring">tensor ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+ring">symmetric ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+ring">differential ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Q-algebra">Q-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordered+ring">ordered ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice-ordered+ring">lattice-ordered ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+ordered+ring">totally ordered ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tight+apartness+ring">tight apartness ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/apartness+ring">apartness ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inequality+ring">inequality ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+ring">topological ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normed+ring">normed ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a>, <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></p> </li> <li> <p>many more…</p> </li> </ul> <h3 id="integral_domains">Integral domains</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GCD+domain">GCD domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B%C3%A9zout+domain">Bézout domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+domain">Euclidean domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unique+factorization+domain">unique factorization domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+domain">principal ideal domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordered+integral+domain">ordered integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/archimedean+integral+domain">archimedean integral domain</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonunital+ring">nonunital ring</a>, <a class="existingWikiWord" href="/nlab/show/nonassociative+ring">nonassociative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/group+object">group object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quadratic+abelian+group">quadratic abelian group</a></p> </li> <li> <p><strong>ring</strong>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subring">subring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+groupoid">ring groupoid</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+ring+groupoid">symmetric ring groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+object">ring object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">localization of a ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+ring">filtered ring</a>, <a class="existingWikiWord" href="/nlab/show/associated+graded+ring">associated graded ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/w-contractible+ring">w-contractible ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/core+of+a+ring">core of a ring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+ring">prime ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+a+commutative+ring">spectrum of a commutative ring</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/affine+scheme">affine scheme</a>, <a class="existingWikiWord" href="/nlab/show/affine+scheme">affine scheme</a>, <a class="existingWikiWord" href="/nlab/show/spectral+topological+space">spectral topological space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+extension">ring extension</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+extension">infinitesimal extension</a></p> <p><a class="existingWikiWord" href="/nlab/show/nilradical">nilradical</a>, <a class="existingWikiWord" href="/nlab/show/reduced+ring">reduced ring</a>,</p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pythagorean+ring">Pythagorean ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/near-ring">near-ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+setoid">ring setoid</a></p> </li> </ul> <h2 id="References">References</h2> <h3 id="ReferencesGeneral">General</h3> <p>Lecture notes:</p> <ul> <li>Arno Fehm, <em>Ringe</em> (<a href="http://www.math.uni-konstanz.de/~fehm/teaching/algebra/geyer2.pdf">pdf</a>) (in German)</li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Frank+W.+Anderson">Frank W. Anderson</a>, <a class="existingWikiWord" href="/nlab/show/Kent+R.+Fuller">Kent R. Fuller</a>, <em>Rings and Categories of Modules</em>, Graduate Texts in Mathematics, <strong>13</strong> Springer (1992) [<a href="https://doi.org/10.1007/978-1-4612-4418-9">doi:10.1007/978-1-4612-4418-9</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Edwin+Bland">Paul Edwin Bland</a>, <em>Rings and Their Modules</em>, De Gruyter (2011) [<a href="https://doi.org/10.1515/9783110250237">doi:10.1515/9783110250237</a>, <a href="http://site.iugaza.edu.ps/mashker/files/%D9%85%D8%AD%D8%A7%D8%B6%D8%B1%D8%A7%D8%AA-%D8%AC%D8%A8%D8%B1-%D8%AF%D9%83%D8%AA%D9%88%D8%B1%D8%A7%D8%A9.pdf">pdf</a>]</p> </li> </ul> <p id="TTFormalizations"> Formalization in <a class="existingWikiWord" href="/nlab/show/univalent+foundations+of+mathematics">univalent foundations of mathematics</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> with the <a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a>):</p> <p>in a context of plain <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mart%C3%ADn+Escard%C3%B3">Martín Escardó</a>, <em><a href="https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#ring-sip">Rings</a></em>, §3.33.13 in: <em>Introduction to Univalent Foundations of Mathematics with Agda</em> [<a href="https://arxiv.org/abs/1911.00580">arXiv:1911.00580</a>, <a href="https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html">webpage</a>]</li> </ul> <p>in a context of <a class="existingWikiWord" href="/nlab/show/cubical+type+theory">cubical</a> <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/1lab">1lab</a>: <em><a href="https://1lab.dev/Algebra.Ring.html">Algebra.Ring</a></em></li> </ul> <h3 id="ReferencesHistory">History</h3> <p><a class="existingWikiWord" href="/nlab/show/Richard+Dedekind">Richard Dedekind</a> had introduced the concept today called <em>ring</em> under the name <em>Ordnung</em> (Ger: order, as in <a href="http://en.wikipedia.org/wiki/Order_%28biology%29">taxonomic order</a>). The word <em>Zahlring</em> (Ger: number ring/ring of numbers) for this was introduced in section 9.31 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+Hilbert">David Hilbert</a>, <em>Die Theorie der algebraischen Zahlkörper</em>, Jahresbericht der Deutschen Mathematiker-Vereinigung 4 (1879)</li> </ul> <p>There, the word ring just appears with a footnote mentioning Dedekind’s use of the word “Ordnung”, no further motivation is given. So probably Hilbert meant to use “ring” as in “collection of things holding together”, not in the sense of circles or loops (as one might guess from the rings of <a class="existingWikiWord" href="/nlab/show/cyclic+groups">cyclic groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_n</annotation></semantics></math>).</p> <p>The first abstract axiomatic description of rings is in</p> <ul> <li>Adolf Fraenkel, Journal für die reine und angewandte Mathematik <strong>145</strong> (1914)</li> </ul> <p>which however contains some additional axioms not used anymore. The set of axioms in its modern form appears first in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Emmy+Noether">Emmy Noether</a>, <em>Ideal Theory in Rings</em>, Mathematische Annalen <strong>83</strong> (1921)</li> </ul> <p>For historical accounts see</p> <ul> <li> <p>I. Kleiner, <em>From numbers to rings: the early history of ring theory</em>, Elemente der Mathematik 53 (1998) 18-35. (<a href="http://dx.doi.org/10.5169/seals-3627">web</a>)</p> </li> <li> <p><em>The development of ring theory</em> (<a href="http://www-history.mcs.st-and.ac.uk/HistTopics/Ring_theory.html">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 11, 2025 at 14:33:45. See the <a href="/nlab/history/ring" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/ring" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4173/#Item_24">Discuss</a><span class="backintime"><a href="/nlab/revision/ring/99" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/ring" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/ring" accesskey="S" class="navlink" id="history" rel="nofollow">History (99 revisions)</a> <a href="/nlab/show/ring/cite" style="color: black">Cite</a> <a href="/nlab/print/ring" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/ring" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>