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In [[abstract algebra]], an '''Artinian ring''' is a [[ring (mathematics)|ring]] that satisfies the [[descending chain condition]] on [[ideal (ring theory)|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 ring]]s and rings that are finite-dimensional [[vector space]]s over [[field (mathematics)|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 ring]]s the left and right definitions coincide, but in general they are distinct from each other.


The [[Artin–Wedderburn theorem]] characterizes all [[simple ring|simple]] Artinian rings as the [[matrix ring]]s over a [[division ring]]. This implies that a simple ring is left Artinian if and only if it is right Artinian.
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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 [[Hopkins–Levitzki theorem|Akizuki–Hopkins–Levitzki theorem]] is that a left (right) Artinian ring is automatically a left (right) [[Noetherian ring]]. This is not true for general [[module (mathematics)|module]]s, 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., <math>\mathbb{Z}/n \mathbb{Z}</math>) is left and right Artinian.
*Let ''k'' be a field. Then <math>k[t]/(t^n)</math> is Artinian for every positive integer ''n''.
*If ''I'' is a nonzero ideal of a [[Dedekind domain]] ''A'', then <math>A/I</math> is a [[principal ideal ring|principal]] Artinian ring.<ref>Theorem 459 of http://math.uga.edu/~pete/integral.pdf</ref>
*For each <math>n \ge 1</math>, the full matrix ring <math>M_n(R)</math> over a left Artinian (resp. left Noetherian) ring ''R'' is left Artinian (resp. left Noetherian).<ref>{{harvnb|Cohn|2003|loc=5.2 Exercise 11}}</ref>
 
The ring of integers <math>\mathbb{Z}</math> 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: (i) ''M'' is finitely generated, (ii) ''M'' has [[finite length]], (iii) ''M'' is Noetherian, (iv) ''M'' is Artinian.<ref>{{harvnb|Bourbaki|loc=VIII, pg 7}}</ref>
 
== 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.<ref>{{Harvard citations|nb = yes|last1 = Atiyah|last2 = Macdonald|year = 1969|}}, Theorems 8.7</ref>
*''A''&nbsp;/&nbsp;nil(''A'') is a [[semisimple ring]], where nil(''A'') is the [[Nilradical of a ring|nilradical]] of ''A''{{citation needed|date=July 2012}}.<ref>Sketch: In commutative rings, nil(''A'') is contained in the [[Jacobson radical]] of ''A''. Since ''A''/nil(''A'') is semisimple, nil(''A'') is actually equal to the Jacobson radical of ''A''. By [[Levitzky's theorem]], nil(''A'') is a nilpotent ideal. These last two facts show that ''A'' is a [[semiprimary ring]], and by the [[Hopkins–Levitzki theorem]] ''A'' is Artinian.</ref>
* Every finitely generated module over ''A'' has finite length. (see above)
*''A'' has dimension zero.<ref>{{Harvard citations|nb = yes|last1 = Atiyah|last2 = Macdonald|year = 1969|}}, Theorems 8.5</ref> (In particular, the nilradical is the Jacobson radical since prime ideals are maximal.)
*<math>\operatorname{Spec}A</math> is finite and discrete.
*<math>\operatorname{Spec}A</math> is discrete.<ref>{{Harvard citations|nb = yes|last1 = Atiyah|last2 = Macdonald|year = 1969|}}, Ch. 8, Exercise 2.</ref>
 
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.
 
==See also==
* [[Artin algebra]]
* [[Artinian ideal]]
* [[Serial module]]
* [[Semiperfect ring]]
* [[Noetherian ring]]
 
==Notes==
<references/>
 
==References==
 
*{{Citation | last1=Auslander | first1=Maurice | last2=Reiten | first2=Idun | last3=Smalø | first3=Sverre O. | title=Representation theory of Artin algebras | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-41134-9 | doi=10.1017/CBO9780511623608 | mr=1314422  | year=1995 | volume=36}}
* Bourbaki, Algèbre
* Charles Hopkins. Rings with minimal condition for left ideals.  Ann. of Math. (2)  40, (1939). 712–730.
*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}}
* {{cite book |last1=Cohn |first1=Paul Moritz |authorlink1= |last2= |first2= |authorlink2= |title=Basic algebra: groups, rings, and fields |url= |edition= |series= |volume= |year=2003 |publisher=Springer |location= |isbn=978-1-85233-587-8 |id= }}
 
[[Category:Ring theory]]

Latest revision as of 10:02, 6 December 2014


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