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href='/nlab/show/monad'>monad</a> / <a class='existingWikiWord' href='/nlab/show/%28infinity%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/%28infinity%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/infinity-algebra+over+an+%28infinity%2C1%29-monad'>∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/algebra+over+a+Lawvere+theory'>algebra over an algebraic theory</a></p> <p><a class='existingWikiWord' href='/nlab/show/infinity-algebra+over+an+%28infinity%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/infinity-algebra+over+an+%28infinity%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/infinity-action'>∞-action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/infinity-representation'>∞-representation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/module'>module</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-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+infinity-bundle'>associated ∞-bundle</a></p> </li> </ul> <h2 id='higher_algebras'>Higher algebras</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/monoidal+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoid in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/commutative+monoid+in+a+symmetric+monoidal+%28infinity%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+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-infinity-algebra'>A-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/A-infinity-ring'>A-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/A-infinity-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-infinity-ring'>E-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/E-infinity+algebra'>E-∞ algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-module'>∞-module</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%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-infinity-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+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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><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><li><a href='#examples'>Examples</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#general'>General</a></li><li><a href='#a_homological_characterization'>A homological characterization</a></li></ul></li><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/nonunital+ring'>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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> has a canonical <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_3' xmlns='http://www.w3.org/1998/Math/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/action'>left action</a> <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_4' xmlns='http://www.w3.org/1998/Math/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/action'>right action</a> <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_5' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_6' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_7' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_9' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_10' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_11' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_12' xmlns='http://www.w3.org/1998/Math/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+in+a+ring'>category of two-sided ideals</a> in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_13' xmlns='http://www.w3.org/1998/Math/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/ideal'>two-sided ideals</a> <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, <a class='existingWikiWord' href='/nlab/show/bimodule'>sub-$R$-$R$-bimodules</a> of <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_16' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, and whose <a class='existingWikiWord' href='/nlab/show/morphism'>morphisms</a> are <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/bimodule'>bimodule monomorphisms</a>.</p> <p>An ascending chain of two-sided ideals in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_20' xmlns='http://www.w3.org/1998/Math/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+limit'>direct sequence</a> of two-sided ideals in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_21' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_22' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-bimodule monomorphisms: for natural number <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_25' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-bimodule monomorphism <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_28' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_29' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_30' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, there exists a natural number <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_32' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_33' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-bimodule monomorphism <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_36' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/bimodule'>bimodule isomorphism</a>.</p> <h3 id='left_noetherian_rings'>Left Noetherian rings</h3> <p>Let <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_39' xmlns='http://www.w3.org/1998/Math/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/ideal'>left ideals</a> in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_40' xmlns='http://www.w3.org/1998/Math/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/ideal'>left ideals</a> <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, sub-left-<math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-modules of <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_44' xmlns='http://www.w3.org/1998/Math/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/module'>left module</a> structure <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_45' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, and whose <a class='existingWikiWord' href='/nlab/show/morphism'>morphisms</a> are left <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_47' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_48' xmlns='http://www.w3.org/1998/Math/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+limit'>direct sequence</a> of left ideals in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_49' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_50' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module monomorphisms: for natural number <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_52' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module monomorphism <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_54' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_55' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_56' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, there exists a natural number <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_58' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_59' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module monomorphism <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_61' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_62' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_63' xmlns='http://www.w3.org/1998/Math/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/ideal'>right ideals</a> in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_64' xmlns='http://www.w3.org/1998/Math/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/ideal'>right ideals</a> <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, sub-right-<math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-modules of <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_68' xmlns='http://www.w3.org/1998/Math/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/module'>right module</a> structure <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_69' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, and whose <a class='existingWikiWord' href='/nlab/show/morphism'>morphisms</a> are right <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_71' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_72' xmlns='http://www.w3.org/1998/Math/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+limit'>direct sequence</a> of right ideals in <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_73' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_74' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module monomorphisms: for natural number <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_76' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module monomorphism <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_78' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_79' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_80' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, there exists a natural number <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_82' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_83' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module monomorphism <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_85' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_86' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_87' xmlns='http://www.w3.org/1998/Math/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'>1</a>) then</p> <ol> <li> <p>the <a class='existingWikiWord' href='/nlab/show/polynomial'>polynomial algebra</a> <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_88' xmlns='http://www.w3.org/1998/Math/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/power+series'>formal power series algebra</a> <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_89' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/coordinate+system'>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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_91' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_92' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_93' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is a left ideal of <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_95' xmlns='http://www.w3.org/1998/Math/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/dependent+choice'>axiom of dependent choice</a>, there is a <a class='existingWikiWord' href='/nlab/show/sequence'>sequence</a> of polynomials <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_96' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_97' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_98' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_99' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_100' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_101' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_102' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_103' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is finitely generated; say <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_105' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_106' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_107' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>I_k</annotation></semantics></math>, so <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_109' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_110' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> has degree <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_112' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_113' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_114' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_115' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_116' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_117' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_118' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> the following are equivalent:</p> <ol> <li><math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_120' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-modules is injective.</li> <li><math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_122' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_123' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_124' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is Noetherian and <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_126' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_127' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔧</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{j}</annotation></semantics></math> any morphism of left modules <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_129' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_130' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is Notherian, <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_132' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> lies in a finite sum <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_134' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> exists by the injectivity of each <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_136' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_137' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_138' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_139' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_140' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_141' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_142' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_143' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>I_k</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_145' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_146' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_147' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>I</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>I_k</annotation></semantics></math>. Therefore, <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_149' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mo>⇒</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>2 \Rightarrow 3</annotation></semantics></math>: <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_151' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_152' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_153' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_154' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is injective iff <math class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_156' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_93cbfb14e8e54213dd72a0baa4cfd1bf9dcb68b8_157' xmlns='http://www.w3.org/1998/Math/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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Noetherian+bimodule'>Noetherian bimodule</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='http://en.wikipedia.org/wiki/Noetherian_ring'>wikipedia</a>, <em><a class='existingWikiWord' href='/nlab/show/noetherian+object'>noetherian object</a></em></p> </li> </ul> <p> </p> </div>  <div class="revisedby"> <p> Revision on May 7, 2023 at 21:58:08 by <a href="/nlab/author/Dmitri+Pavlov" style="color: #005c19">Dmitri Pavlov</a> See the <a href="/nlab/history/noetherian+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="https://nforum.ncatlab.org/discussion/7043/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/show/noetherian+ring" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (to current)</span><span class="backintime"><a href="/nlab/revision/noetherian+ring/21" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (21 more)</span><a href="/nlab/show/noetherian+ring" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/noetherian+ring/22" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/noetherian+ring" accesskey="S" class="navlink" id="history" rel="nofollow">History (22 revisions)</a><a href="/nlab/rollback/noetherian+ring?rev=22" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/noetherian+ring/22/cite" style="color: black">Cite</a> <a href="/nlab/source/noetherian+ring/22" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>