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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> </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> <ul> <li><a href='#noetherian_rings'>Noetherian rings</a></li> <li><a href='#left_noetherian_rings'>Left Noetherian rings</a></li> <li><a href='#right_noetherian_rings'>Right Noetherian rings</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#a_homological_characterization'>A homological characterization</a></li> </ul> <li><a href='#noetherian_and_artinian_rings'>Noetherian and Artinian rings</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A Noetherian (or often, as below, noetherian) <a class="existingWikiWord" href="/nlab/show/ring">ring</a> (or <a class="existingWikiWord" href="/nlab/show/rng">rng</a>) is one where it is possible to do <a class="existingWikiWord" href="/nlab/show/induction">induction</a> over its ideals, because the ordering of ideals by reverse inclusion is <a class="existingWikiWord" href="/nlab/show/well-founded+relation">well-founded</a>.</p> <h2 id="definition">Definition</h2> <h3 id="noetherian_rings">Noetherian rings</h3> <p>Every <a class="existingWikiWord" href="/nlab/show/ring">ring</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> has a canonical <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>-<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/bimodule">bimodule</a> structure, with <a class="existingWikiWord" href="/nlab/show/left+action">left action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>L</mi></msub><mo>:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\alpha_L:R \times R \to R</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/right+action">right action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo>:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\alpha_R:R \times R \to R</annotation></semantics></math> defined as the multiplicative binary operation on <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> and <a class="existingWikiWord" href="/nlab/show/biaction">biaction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\alpha:R \times R \times R \to R</annotation></semantics></math> defined as the ternary product on <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>L</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>a</mi><mo>⋅</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\alpha_L(a, b) \coloneqq a \cdot b</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>a</mi><mo>⋅</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\alpha_R(a, b) \coloneqq a \cdot b</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>⋅</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\alpha(a, b, c) \coloneqq a \cdot b \cdot c</annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">TwoSidedIdeals</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{TwoSidedIdeals}(R)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/category+of+two-sided+ideals">category of two-sided ideals</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>, whose objects are <a class="existingWikiWord" href="/nlab/show/two-sided+ideals">two-sided ideals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> 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>, <a class="existingWikiWord" href="/nlab/show/subbimodules">sub-<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>-<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>-bimodules</a> of <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> with respect to the canonical bimodule structure on <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>, and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <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>-<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/bimodule+monomorphisms">bimodule monomorphisms</a>.</p> <p>An ascending chain of two-sided ideals 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 <a class="existingWikiWord" href="/nlab/show/direct+sequence">direct sequence</a> of two-sided ideals 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>, a <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> of two-sided ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi mathvariant="normal">TwoSidedIdeals</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A:\mathbb{N} \to \mathrm{TwoSidedIdeals}(R)</annotation></semantics></math> with the following <a class="existingWikiWord" href="/nlab/show/dependent+sequence">dependent sequence</a> of <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>-<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>-bimodule monomorphisms: for natural number <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>, a dependent <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>-<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>-bimodule monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">i_n:A_{n} \hookrightarrow A_{n+1}</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/ring">ring</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> is <strong>Noetherian</strong> if it satisfies the <a class="existingWikiWord" href="/nlab/show/ascending+chain+condition">ascending chain condition</a> on its two-sided ideals: for every ascending chain of two-sided ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, i_n)</annotation></semantics></math> 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>, there exists a natural number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">m \in \mathbb{N}</annotation></semantics></math> such that for all natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n \geq m</annotation></semantics></math>, the <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>-<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>-bimodule monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">i_n:A_{n} \hookrightarrow A_{n+1}</annotation></semantics></math> is an <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>-<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/bimodule+isomorphism">bimodule isomorphism</a>.</p> <h3 id="left_noetherian_rings">Left Noetherian rings</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">LeftIdeals</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{LeftIdeals}(R)</annotation></semantics></math> be the category of <a class="existingWikiWord" href="/nlab/show/left+ideals">left ideals</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>, whose objects are <a class="existingWikiWord" href="/nlab/show/left+ideals">left ideals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> 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>, sub-left-<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>-modules of <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> with respect to the canonical <a class="existingWikiWord" href="/nlab/show/left+module">left module</a> structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">(-)\cdot(-):R \times R \to R</annotation></semantics></math> on <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>, and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are left <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>-module monomorphisms.</p> <p>An ascending chain of left ideals 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 <a class="existingWikiWord" href="/nlab/show/direct+sequence">direct sequence</a> of left ideals 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>, a <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> of left ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi mathvariant="normal">LeftIdeals</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A:\mathbb{N} \to \mathrm{LeftIdeals}(R)</annotation></semantics></math> with the following <a class="existingWikiWord" href="/nlab/show/dependent+sequence">dependent sequence</a> of left <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>-module monomorphisms: for natural number <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>, a dependent left <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>-module monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">i_n:A_{n} \hookrightarrow A_{n+1}</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/ring">ring</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> is <strong>left Noetherian</strong> if it satisfies the <a class="existingWikiWord" href="/nlab/show/ascending+chain+condition">ascending chain condition</a> on its left ideals: for every ascending chain of left ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, i_n)</annotation></semantics></math> 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>, there exists a natural number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">m \in \mathbb{N}</annotation></semantics></math> such that for all natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n \geq m</annotation></semantics></math>, the left <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>-module monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">i_n:A_{n} \hookrightarrow A_{n+1}</annotation></semantics></math> is an left <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>-module isomorphism.</p> <h3 id="right_noetherian_rings">Right Noetherian rings</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">RightIdeals</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{RightIdeals}(R)</annotation></semantics></math> be the category of <a class="existingWikiWord" href="/nlab/show/right+ideals">right ideals</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>, whose objects are <a class="existingWikiWord" href="/nlab/show/right+ideals">right ideals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> 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>, sub-right-<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>-modules of <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> with respect to the canonical <a class="existingWikiWord" href="/nlab/show/right+module">right module</a> structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">(-)\cdot(-):R \times R \to R</annotation></semantics></math> on <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>, and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are right <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>-module monomorphisms.</p> <p>An ascending chain of right ideals 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 <a class="existingWikiWord" href="/nlab/show/direct+sequence">direct sequence</a> of right ideals 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>, a <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> of right ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi mathvariant="normal">RightIdeals</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A:\mathbb{N} \to \mathrm{RightIdeals}(R)</annotation></semantics></math> with the following <a class="existingWikiWord" href="/nlab/show/dependent+sequence">dependent sequence</a> of right <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>-module monomorphisms: for natural number <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>, a dependent right <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>-module monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">i_n:A_{n} \hookrightarrow A_{n+1}</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/ring">ring</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> is <strong>right Noetherian</strong> if it satisfies the <a class="existingWikiWord" href="/nlab/show/ascending+chain+condition">ascending chain condition</a> on its right ideals: for every ascending chain of right ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, i_n)</annotation></semantics></math> 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>, there exists a natural number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">m \in \mathbb{N}</annotation></semantics></math> such that for all natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n \geq m</annotation></semantics></math>, the right <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>-module monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">i_n:A_{n} \hookrightarrow A_{n+1}</annotation></semantics></math> is an right <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>-module isomorphism.</p> <h2 id="examples">Examples</h2> <div class="num_example" id="FieldIsNoetherianRing"> <h6 id="example">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/field">field</a> is a noetherian ring.</p> </div> <div class="num_example" id="PIDIsNoetherianRing"> <h6 id="example_2">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/principal+ideal+domain">principal ideal domain</a> is a noetherian ring.</p> </div> <div class="num_example" id="PolynomialAlgebraOverNoetherianRingIsNoetherian"> <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 <a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a> (e.g. a <a class="existingWikiWord" href="/nlab/show/field">field</a> by example <a class="maruku-ref" href="#FieldIsNoetherianRing"></a>) then</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/polynomial+algebra">polynomial algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R[X_1, \cdots, X_n]</annotation></semantics></math></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/formal+power+series+algebra">formal power series algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R[ [ X_1, \cdots, X_n ] ]</annotation></semantics></math></p> </li> </ol> <p>over R in a <a class="existingWikiWord" href="/nlab/show/finite+number">finite number</a> <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> of <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> are Noetherian.</p> </div> <h2 id="properties">Properties</h2> <p>Spectra of noetherian rings are glued together to define <a class="existingWikiWord" href="/nlab/show/noetherian+scheme">locally noetherian schemes</a>.</p> <h3 id="general">General</h3> <p>One of the best-known properties is the Hilbert basis theorem. Let <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> be a (unital) ring.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Hilbert)</strong> If <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 left Noetherian, then so is the <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial algebra</a> <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>. (Similarly if “right” is substituted for “left”.)</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>(We adapt the proof from <a href="https://en.wikipedia.org/wiki/Hilbert%27s_basis_theorem#First_Proof">Wikipedia</a>.) Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a left ideal of <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> that is not finitely generated. Using the <a class="existingWikiWord" href="/nlab/show/axiom+of+dependent+choice">axiom of dependent choice</a>, there is a <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> of polynomials <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f_n \in I</annotation></semantics></math> such that the left ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>n</mi></msub><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>0</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I_n \coloneqq (f_0, \ldots, f_{n-1})</annotation></semantics></math> form a strictly increasing chain and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub><mo>∈</mo><mi>I</mi><mo>∖</mo><msub><mi>I</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_n \in I \setminus I_n</annotation></semantics></math> is chosen to have degree as small as possible. Putting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mo>≔</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_n \coloneqq \deg(f_n)</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo>≤</mo><msub><mi>d</mi> <mn>1</mn></msub><mo>≤</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">d_0 \leq d_1 \leq \ldots</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">a_n</annotation></semantics></math> be the leading coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_n</annotation></semantics></math>. The left ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_0, a_1, \ldots)</annotation></semantics></math> of <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 finitely generated; say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>a</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_0, \ldots, a_{k-1})</annotation></semantics></math> generates. Thus we may write</p> <div class="maruku-equation" id="eq:kill"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>k</mi></msub><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mi>r</mi> <mi>i</mi></msub><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> a_k = \sum_{i=0}^{k-1} r_i a_i</annotation></semantics></math></div> <p>The polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi>r</mi> <mi>i</mi></msub><msup><mi>x</mi> <mrow><msub><mi>d</mi> <mi>k</mi></msub><mo>−</mo><msub><mi>d</mi> <mi>i</mi></msub></mrow></msup><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">g = \sum_{i=0}^{k-1} r_i x^{d_k - d_i} f_i</annotation></semantics></math> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">I_k</annotation></semantics></math>, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub><mo>−</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f_k - g</annotation></semantics></math> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∖</mo><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">I \setminus I_k</annotation></semantics></math>. Also <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> has degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math> or less, and therefore so does <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub><mo>−</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f_k - g</annotation></semantics></math>. But notice that the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mrow><msub><mi>d</mi> <mi>k</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">x^{d_k}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub><mo>−</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f_k - g</annotation></semantics></math> is zero, by <a class="maruku-eqref" href="#eq:kill">(1)</a>. So in fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub><mo>−</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f_k - g</annotation></semantics></math> has degree less than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math>, contradicting how <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math> was chosen.</p> </div> <h3 id="a_homological_characterization">A homological characterization</h3> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>For a unital ring <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> the following are equivalent:</p> <ol> <li><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 left Noetherian</li> <li>Any small direct sum of injective left <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>-modules is injective.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Ext</mo> <mi>R</mi> <mi>k</mi></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\operatorname{Ext}^k_R(A, \cdot)</annotation></semantics></math> commutes with small direct sums for any finitely generated <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>.</li> </ol> </div> <p>Direct sums here can be replaced by filtered colimits.</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>⇒</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">1 \Rightarrow 2</annotation></semantics></math>: assume that <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 Noetherian and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">I_\alpha</annotation></semantics></math> are injective modules. In order to verify that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>:</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mi>α</mi></msub><msub><mi>I</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">I := \bigoplus_\alpha I_\alpha</annotation></semantics></math> is injective it is enough to show that for any ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔧</mi></mrow><annotation encoding="application/x-tex">\mathfrak{j}</annotation></semantics></math> any morphism of left modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>𝔧</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f : \mathfrak{j} \to I</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔧</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathfrak{j} \to R</annotation></semantics></math>. Since <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 Notherian, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔧</mi></mrow><annotation encoding="application/x-tex">\mathfrak{j}</annotation></semantics></math> is finitely generated, so the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> lies in a finite sum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mrow><msub><mi>α</mi> <mn>1</mn></msub></mrow></msub><mo>⊕</mo><mi>…</mi><mo>⊕</mo><msub><mi>I</mi> <mrow><msub><mi>α</mi> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">I_{\alpha_1} \oplus \dots \oplus I_{\alpha_n}</annotation></semantics></math>. Thus an extension to <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> exists by the injectivity of each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mrow><msub><mi>α</mi> <mi>k</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">I_{\alpha_k}</annotation></semantics></math>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo>⇒</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2 \Rightarrow 1</annotation></semantics></math>: if <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 not left Noetherian then there is a sequence of left ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔧</mi> <mn>1</mn></msub><mo>⊊</mo><msub><mi>𝔧</mi> <mn>2</mn></msub><mo>⊊</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">\mathfrak{j}_1 \subsetneq \mathfrak{j}_2 \subsetneq \dots</annotation></semantics></math>. Take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔧</mi><mo>:</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo> <mi>k</mi></msub><msub><mi>𝔧</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{j} := \bigcup_k \mathfrak{j}_k</annotation></semantics></math>. The obvious map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>𝔧</mi><mo stretchy="false">/</mo><msub><mi>𝔧</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j \to \prod_k (\mathfrak{j} / \mathfrak{j}_k)</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>𝔧</mi><mo stretchy="false">/</mo><msub><mi>𝔧</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\bigoplus_k (\mathfrak{j} / \mathfrak{j}_k)</annotation></semantics></math>, since any element lies in all but finitely many <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔧</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{j}_k</annotation></semantics></math>. Now take any injective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">I_k</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>𝔧</mi><mo stretchy="false">/</mo><msub><mi>𝔧</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">0 \to \mathfrak{j} / \mathfrak{j}_k \to I_k</annotation></semantics></math>. The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔧</mi><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mi>k</mi></msub><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{j} \to \bigoplus_k I_k</annotation></semantics></math> cannot extend to the whole <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>, since otherwise its image would be contained in a sum of finitely many <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">I_k</annotation></semantics></math>. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mi>k</mi></msub><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\bigoplus_k I_k</annotation></semantics></math> is not injective.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo>⇒</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">2 \Rightarrow 3</annotation></semantics></math>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Ext</mo> <mi>R</mi> <mi>k</mi></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mi>α</mi></msub><msub><mi>X</mi> <mi>α</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\operatorname{Ext}^k_R(A, \bigoplus_\alpha X_\alpha)</annotation></semantics></math> can be computed by taking an injective resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mi>α</mi></msub><msub><mi>X</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">\bigoplus_\alpha X_\alpha</annotation></semantics></math>. Since direct sums of injective modules are assumed to be injective, we can take a direct sum of injective resolutions of each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">X_\alpha</annotation></semantics></math>. It remains to note that Hom out of a finitely generated module commutes with arbitrary direct sums.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn><mo>⇒</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">3 \Rightarrow 2</annotation></semantics></math>: Follows from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is injective iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Ext</mo> <mi>R</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">/</mo><mi>𝔦</mi><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\operatorname{Ext}^1_R(R / \mathfrak{i}, I) = 0</annotation></semantics></math> for any ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔦</mi></mrow><annotation encoding="application/x-tex">\mathfrak{i}</annotation></semantics></math>.</p> </div> <h2 id="noetherian_and_artinian_rings">Noetherian and Artinian rings</h2> <p>A dual condition is artinian: an <strong><a class="existingWikiWord" href="/nlab/show/artinian+ring">artinian ring</a></strong> is a ring satisfying the descending chain condition on ideals. The symmetry is severely broken if one considers unital rings: for example every unital artinian ring is noetherian; artinian rings are intuitively much smaller than generic noetherian rings.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+module">Noetherian module</a>, <a class="existingWikiWord" href="/nlab/show/Noetherian+bimodule">Noetherian bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noetherian+object">noetherian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+poset">Noetherian poset</a></p> </li> <li> <p><span class="newWikiWord">Noetherian E-∞ ring<a href="/nlab/new/Noetherian+E-%E2%88%9E+ring">?</a></span></p> </li> </ul> <h2 id="references">References</h2> <p>Introduced by <a class="existingWikiWord" href="/nlab/show/Emmy+Noether">Emmy Noether</a> in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Emmy+Noether">Emmy Noether</a>, <em>Idealtheorie in Ringbereichen</em>, Mathematische Annalen 83:1 (1921), 24–66. <a href="https://doi.org/10.1007/bf01464225">doi:10.1007/bf01464225</a>.</p> </li> <li> <p><a href="https://en.wikipedia.org/wiki/Noetherian_ring">wikipedia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K.+R.+Goodearl">K. R. Goodearl</a>, R. B. Warfield, <em>An introduction to noncommutative Noetherian rings</em>, London Math. Society Student Texts <strong>16</strong> (1st ed,), 1989, xviii+303 pp.; or 61 (2nd ed.), 2004, xxiv+344 pp.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 12, 2024 at 17:51:19. 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