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In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.

A ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other.

The Artin–Wedderburn theorem characterizes all simple Artinian rings as the matrix rings over a division ring. This implies that a simple ring is left Artinian if and only if it is right Artinian.

Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition. Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem is that a left (resp. right) Artinian ring is automatically a left (resp. right) Noetherian ring. This is not true for general modules, that is, an Artinian module need not be a Noetherian module.

Examples

• An integral domain is Artinian if and only if it is a field.


• A ring with finitely many, say left, ideals is left Artinian. In particular, a finite ring (e.g., Z/nZ{displaystyle mathbb {Z} /nmathbb {Z} }) is left and right Artinian.


• Let k be a field. Then k[t]/(tn){displaystyle k[t]/(t^{n})} is Artinian for every positive integer n.


• Similarly, k[x,y]/(x2,y3,xy2)=k⊕k⋅x⊕k⋅y⊕k⋅xy⊕k⋅y2{displaystyle k[x,y]/(x^{2},y^{3},xy^{2})=koplus kcdot xoplus kcdot yoplus kcdot xyoplus kcdot y^{2}} is an Artinian ring with maximal ideal (x,y){displaystyle (x,y)}
  


• If I is a nonzero ideal of a Dedekind domain A, then A/I{displaystyle A/I} is a principal Artinian ring.^[1]
  


• For each n≥1{displaystyle ngeq 1}, the full matrix ring Mn(R){displaystyle M_{n}(R)} over a left Artinian (resp. left Noetherian) ring R is left Artinian (resp. left Noetherian).^[2]
  

The ring of integers Z{displaystyle mathbb {Z} } is a Noetherian ring but is not Artinian.

Modules over Artinian rings

Let M be a left module over a left Artinian ring. Then the following are equivalent (Hopkins' theorem): (i) M is finitely generated, (ii) M has finite length (i.e., has composition series), (iii) M is Noetherian, (iv) M is Artinian.^[3]

Commutative Artinian rings

Let A be a commutative Noetherian ring with unity. Then the following are equivalent.

• 
  A is Artinian.


• 
  A is a finite product of commutative Artinian local rings.^[4]
  


• 
  A / nil(A) is a semisimple ring, where nil(A) is the nilradical of A.^{[citation needed]}
  


• Every finitely generated module over A has finite length. (see above)


• 
  A has Krull dimension zero.^[5] (In particular, the nilradical is the Jacobson radical since prime ideals are maximal.)


• 
  Spec⁡A{displaystyle operatorname {Spec} A} is finite and discrete.


• 
  Spec⁡A{displaystyle operatorname {Spec} A} is discrete.^[6]
  

Let k be a field and A finitely generated k-algebra. Then A is Artinian if and only if A is finitely generated as k-module.

An Artinian local ring is complete. A quotient and localization of an Artinian ring is Artinian.

Simple Artinian ring

A simple Artinian ring A is a matrix ring over a division ring. Indeed,^[7] let I be a minimal (nonzero) right ideal of A. Then, since AI{displaystyle AI} is a two-sided ideal, AI=A{displaystyle AI=A} since A is simple. Thus, we can choose ai∈A{displaystyle a_{i}in A} so that 1∈a1I+⋯+akI{displaystyle 1in a_{1}I+cdots +a_{k}I}. Assume k is minimal with respect that property. Consider the map of right A-modules:

{I⊕k→A,(y1,…,yk)↦a1y1+⋯+akyk{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}}}

It is surjective. If it is not injective, then, say, a1y1=a2y2+⋯+akyk{displaystyle a_{1}y_{1}=a_{2}y_{2}+cdots +a_{k}y_{k}} with nonzero y1{displaystyle y_{1}}. Then, by the minimality of I, we have: y1A=I{displaystyle y_{1}A=I}. It follows:

a1I=a1y1A⊂a2I+⋯+akI{displaystyle a_{1}I=a_{1}y_{1}Asubset 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≃EndA⁡(A)≃Mk(EndA⁡(I)){displaystyle Asimeq operatorname {End} _{A}(A)simeq M_{k}(operatorname {End} _{A}(I))}.

See also

• Artin algebra


• Artinian ideal


• Serial module


• Semiperfect ring


• Noetherian ring

Notes

References

• 
  Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1995), Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, doi:10.1017/CBO9780511623608, ISBN 978-0-521-41134-9, MR 1314422
  


• Bourbaki, Algèbre


• Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712–730.


• 
  Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
  


• 
  Cohn, Paul Moritz (2003). Basic algebra: groups, rings, and fields. Springer. ISBN 978-1-85233-587-8.