Artinian ring

<h2 id="examples-and-counterexamples">Examples and counterexamples</h2>
<ul><li>An <a href="/facts/Integral_domain/cylt6zxW">integral domain</a> is Artinian if and only if it is a field.</li>
<li>A ring with finitely many, say left, ideals is left Artinian. In particular, a <a href="/facts/Finite_ring/9paodazt">finite ring</a> (e.g., 
 
 
 
 
 Z
 
 
 /
 
 n
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /n\mathbb {Z} }
 
) is left and right Artinian.</li>
<li>Let k be a field. Then 
 
 
 
 k
 [
 t
 ]
 
 /
 
 (
 
 t
 
 n
 
 
 )
 
 
 {\displaystyle k[t]/(t^{n})}
 
 is Artinian for every positive <a href="/facts/Integer/PUwwrawx">integer</a> n.</li>
<li>Similarly, 
 
 
 
 k
 [
 x
 ,
 y
 ]
 
 /
 
 (
 
 x
 
 2
 
 
 ,
 
 y
 
 3
 
 
 ,
 x
 
 y
 
 2
 
 
 )
 =
 k
 ⊕
 k
 ⋅
 x
 ⊕
 k
 ⋅
 y
 ⊕
 k
 ⋅
 x
 y
 ⊕
 k
 ⋅
 
 y
 
 2
 
 
 
 
 {\displaystyle k[x,y]/(x^{2},y^{3},xy^{2})=k\oplus k\cdot x\oplus k\cdot y\oplus k\cdot xy\oplus k\cdot y^{2}}
 
 is an Artinian ring with <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideal</a> 
 
 
 
 (
 x
 ,
 y
 )
 
 
 {\displaystyle (x,y)}
 
.</li>
<li>Let 
 
 
 
 x
 
 
 {\displaystyle x}
 
 be an endomorphism between a finite-dimensional vector space V. Then the subalgebra 
 
 
 
 A
 ⊂
 End
 ⁡
 (
 V
 )
 
 
 {\displaystyle A\subset \operatorname {End} (V)}
 
 generated by 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is a commutative Artinian ring.</li>
<li>If I is a <a href="/facts/Zero_ideal/PvH2qhxh">nonzero</a> ideal of a <a href="/facts/Dedekind_domain/tAsvAcaW">Dedekind domain</a> A, then 
 
 
 
 A
 
 /
 
 I
 
 
 {\displaystyle A/I}
 
 is a <a href="/facts/Principal_ideal_ring/uvv74rRb">principal</a> Artinian ring.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li>
<li>For each 
 
 
 
 n
 ≥
 1
 
 
 {\displaystyle n\geq 1}
 
, the full <a href="/facts/Matrix_ring/oj5HBr8B">matrix ring</a> 
 
 
 
 
 M
 
 n
 
 
 (
 R
 )
 
 
 {\displaystyle M_{n}(R)}
 
 over a left Artinian (resp. left Noetherian) ring R is left Artinian (resp. left Noetherian).<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li></ul>
The following two are examples of non-Artinian rings.

<ul><li>If R is any ring, then the <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> R[x] is not Artinian, since the ideal generated by 
 
 
 
 
 x
 
 n
 +
 1
 
 
 
 
 {\displaystyle x^{n+1}}
 
 is (properly) contained in the ideal generated by 
 
 
 
 
 x
 
 n
 
 
 
 
 {\displaystyle x^{n}}
 
 for all <a href="/facts/Natural_numbers/u65og8JE">natural numbers</a> n. In contrast, if R is Noetherian so is R[x] by the <a href="/facts/Hilbert_basis_theorem/6Pman9ee">Hilbert basis theorem</a>.</li>
<li>The ring of integers 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 is a Noetherian ring but is not Artinian.</li></ul>
<h2 id="modules-over-artinian-rings">Modules over Artinian rings</h2>
Let M be a left module over a left Artinian ring. Then the following are equivalent (<a href="/facts/Hopkins%2527_theorem/Hh5o9vtm">Hopkins' theorem</a>): (i) M is <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a>, (ii) M has <a href="/facts/Finite_length/UpWodT4k">finite length</a> (i.e., has <a href="/facts/Composition_series/3C7CHH3k">composition series</a>), (iii) M is Noetherian, (iv) M is Artinian.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="commutative-artinian-rings">Commutative Artinian rings</h2>
Let A be a commutative Noetherian ring with unity. Then the following are equivalent.

<ul><li>A is Artinian.</li>
<li>A is a finite <a href="/facts/Product_of_rings/wBGSEE54">product</a> of commutative Artinian <a href="/facts/Local_ring/gRTdg5Qx">local rings</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>A / nil(A) is a <a href="/facts/Semisimple_ring/9KASYk35">semisimple ring</a>, where nil(A) is the <a href="/facts/Nilradical_of_a_ring/7I3pIVeK">nilradical</a> of A.</li>
<li>Every finitely generated module over A has finite length. (see above)</li>
<li>A has <a href="/facts/Krull_dimension/KQb2uLtz">Krull dimension</a> zero.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> (In particular, the nilradical is the <a href="/facts/Jacobson_radical/tTTfKySl">Jacobson radical</a> since <a href="/facts/Prime_ideal/pmrGf6DA">prime ideals</a> are maximal.)</li>
<li>
 
 
 
 Spec
 ⁡
 A
 
 
 {\displaystyle \operatorname {Spec} A}
 
 is finite and discrete.</li>
<li>
 
 
 
 Spec
 ⁡
 A
 
 
 {\displaystyle \operatorname {Spec} A}
 
 is discrete.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li></ul>
Let k be a field and A a finitely generated k-<a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a>. Then A is Artinian if and only if A is finitely generated as a k-module.
An Artinian local ring is complete. A <a href="/facts/Quotient_ring/X2WkXyEw">quotient</a> and <a href="/facts/Localization_(commutative_algebra)/rjevzBJf">localization</a> of an Artinian ring is Artinian.

<h2 id="simple-artinian-ring">Simple Artinian ring</h2>
One version of the <a href="/facts/Wedderburn%25E2%2580%2593Artin_theorem/EB5rY5sd">Wedderburn–Artin theorem</a> states that a <a href="/facts/Simple_ring/5ndCExsb">simple</a> Artinian ring A is a matrix ring over a division ring. Indeed,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> let I be a minimal (nonzero) right ideal of A, which exists since A is Artinian (and the rest of the proof does not use the fact that A is Artinian). Then, since 
 
 
 
 A
 I
 
 
 {\displaystyle AI}
 
 is a two-sided ideal, 
 
 
 
 A
 I
 =
 A
 
 
 {\displaystyle AI=A}
 
 since A is simple. Thus, we can choose 
 
 
 
 
 a
 
 i
 
 
 ∈
 A
 
 
 {\displaystyle a_{i}\in A}
 
 so that 
 
 
 
 1
 ∈
 
 a
 
 1
 
 
 I
 +
 ⋯
 +
 
 a
 
 k
 
 
 I
 
 
 {\displaystyle 1\in a_{1}I+\cdots +a_{k}I}
 
. Assume k is minimal with respect to that property. 
Now consider the map of right A-modules:

{
            
              
                
                  
                    I
                    
                      ⊕
                      k
                    
                  
                  →
                  A
                  ,
                
              
              
                
                  (
                  
                    y
                    
                      1
                    
                  
                  ,
                  …
                  ,
                  
                    y
                    
                      k
                    
                  
                  )
                  ↦
                  
                    a
                    
                      1
                    
                  
                  
                    y
                    
                      1
                    
                  
                  +
                  ⋯
                  +
                  
                    a
                    
                      k
                    
                  
                  
                    y
                    
                      k
                    
                  
                
              
            
            
          
        
      
    
    {\displaystyle {\begin{cases}I^{\oplus k}\to A,\\(y_{1},\dots ,y_{k})\mapsto a_{1}y_{1}+\cdots +a_{k}y_{k}\end{cases}}}

This map is <a href="/facts/Surjective/KezhLpCb">surjective</a>, since the image is a right ideal and contains 1. If it is not <a href="/facts/Injective/5ES6tqf5">injective</a>, then, say, 
 
 
 
 
 a
 
 1
 
 
 
 y
 
 1
 
 
 =
 
 a
 
 2
 
 
 
 y
 
 2
 
 
 +
 ⋯
 +
 
 a
 
 k
 
 
 
 y
 
 k
 
 
 
 
 {\displaystyle a_{1}y_{1}=a_{2}y_{2}+\cdots +a_{k}y_{k}}
 
 with nonzero 
 
 
 
 
 y
 
 1
 
 
 
 
 {\displaystyle y_{1}}
 
. Then, by the minimality of I, we have 
 
 
 
 
 y
 
 1
 
 
 A
 =
 I
 
 
 {\displaystyle y_{1}A=I}
 
. It follows:

a
 
 1
 
 
 I
 =
 
 a
 
 1
 
 
 
 y
 
 1
 
 
 A
 ⊂
 
 a
 
 2
 
 
 I
 +
 ⋯
 +
 
 a
 
 k
 
 
 I
 
 
 {\displaystyle a_{1}I=a_{1}y_{1}A\subset a_{2}I+\cdots +a_{k}I}
 
,
which contradicts the minimality of k. Hence, 
 
 
 
 
 I
 
 ⊕
 k
 
 
 ≃
 A
 
 
 {\displaystyle I^{\oplus k}\simeq A}
 
 and thus 
 
 
 
 A
 ≃
 
 End
 
 A
 
 
 ⁡
 (
 A
 )
 ≃
 
 M
 
 k
 
 
 (
 
 End
 
 A
 
 
 ⁡
 (
 I
 )
 )
 
 
 {\displaystyle A\simeq \operatorname {End} _{A}(A)\simeq M_{k}(\operatorname {End} _{A}(I))}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Artin_algebra/pxuPI9Ca">Artin algebra</a></li>
<li><a href="/facts/Artinian_ideal/M6lA8uw9">Artinian ideal</a></li>
<li><a href="/facts/Serial_module/hB2LGmBG">Serial module</a></li>
<li><a href="/facts/Semiperfect_ring/kvcwAEAF">Semiperfect ring</a></li>
<li><a href="/facts/Gorenstein_ring/D9bw8ocr">Gorenstein ring</a></li>
<li><a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian ring</a></li></ul>
<h2 id="citations">Citations</h2>

<ul><li>Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1995), Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9780511623608">10.1017/CBO9780511623608</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-41134-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1314422">1314422</a></li>
<li>Bourbaki, Nicolas (2012). Algèbre. Chapitre 8, Modules et anneaux semi-simples. Heidelberg: Springer-Verlag Berlin Heidelberg. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-35315-7.</li>
<li>Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712–730.</li>
<li><a href="/facts/Michael_Atiyah/RQVJ9ATE">Atiyah, Michael Francis</a>; <a href="/facts/Ian_G._Macdonald/PoaeNbeg">Macdonald, I.G.</a> (1969), Introduction to Commutative Algebra, Westview Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-201-40751-8</li>
<li>Cohn, Paul Moritz (2003). Basic algebra: groups, rings, and fields. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-85233-587-8.</li>
<li>Brešar, Matej (2014). Introduction to Noncommutative Algebra. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-08692-7.</li>
<li>Clark, Pete L. <a href="https://web.archive.org/web/20101214214852/http://math.uga.edu/~pete/integral.pdf">"Commutative Algebra"</a> (PDF). Archived from <a href="http://math.uga.edu/~pete/integral.pdf">the original</a> (PDF) on 2010-12-14.</li>
<li><a href="/facts/John_Milnor/LTf24paN">Milnor, John Willard</a> (1971), Introduction to algebraic K-theory, Annals of Mathematics Studies, vol. 72, Princeton, NJ: <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0349811">0349811</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0237.18005">0237.18005</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Brešar 2014, p. 73 - Brešar, Matej (2014). Introduction to Noncommutative Algebra. Springer. ISBN 978-3-319-08692-7. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Clark, Theorem 20.11 - Clark, Pete L. "Commutative Algebra" (PDF). Archived from the original (PDF) on 2010-12-14. <a href="https://web.archive.org/web/20101214214852/http://math.uga.edu/~pete/integral.pdf" target="_blank">https://web.archive.org/web/20101214214852/http://math.uga.edu/~pete/integral.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Cohn 2003, 5.2 Exercise 11 - Cohn, Paul Moritz (2003). Basic algebra: groups, rings, and fields. Springer. ISBN 978-1-85233-587-8. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Bourbaki 2012, VIII, p. 7 - Bourbaki, Nicolas (2012). Algèbre. Chapitre 8, Modules et anneaux semi-simples. Heidelberg: Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-35315-7. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Atiyah & Macdonald 1969, Theorems 8.7 - Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Atiyah & Macdonald 1969, Theorems 8.5 - Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Atiyah & Macdonald 1969, Ch. 8, Exercise 2 - Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Milnor 1971, p. 144 - Milnor, John Willard (1971), Introduction to algebraic K-theory, Annals of Mathematics Studies, vol. 72, Princeton, NJ: Princeton University Press, MR 0349811, Zbl 0237.18005 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0349811" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0349811</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Artinian ring open-in-new

Artinian ring