Artinian ring

In abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on ideals; that is, there is no infinite descending sequence of ideals. Artinian rings 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 every simple Artinian ring as a ring of matrices over a division ring. This implies that a simple ring is left Artinian if and only if it is right Artinian.
The same definition and terminology can be applied to modules, with ideals replaced by submodules.
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.
The ring of integers formula_8 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.
Commutative Artinian rings.
Let "A" be a commutative Noetherian ring with unity. Then the following are equivalent.
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, let "I" be a minimal (nonzero) right ideal of "A". Then, since formula_11 is a two-sided ideal, formula_12 since "A" is simple. Thus, we can choose formula_13 so that formula_14. Assume "k" is minimal with respect that property. Consider the map of right "A"-modules:
It is surjective. If it is not injective, then, say, formula_16 with nonzero formula_17. Then, by the minimality of "I", we have: formula_18. It follows:
which contradicts the minimality of "k". Hence, formula_20 and thus formula_21.