In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) 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.
Precisely, 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.
[1] For commutative rings the left and right definitions coincide, but in general they are distinct from each other.
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.
The following two are examples of non-Artinian rings.
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.
[4] Let A be a commutative Noetherian ring with unity.
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.
Indeed,[8] 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).
Assume k is minimal with respect that property.