Artinian ring
In mathematics, specifically abstract algebra, an Artinian 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.
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. For commutative rings the left and right definitions coincide, but in general they are distinct from each other.
The Wedderburn–Artin 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 Artinian ring is automatically a left Noetherian ring. This is not true for general modules; that is, an Artinian module need not be a Noetherian module.
Examples and counterexamples
- 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 is left and right Artinian.
- Let be a field. Then is Artinian for every positive integer n.
- Similarly, is an Artinian ring with maximal ideal.
- Let be an endomorphism between a finite-dimensional vector space V. Then the subalgebra generated by is a commutative Artinian ring.
- If I is a nonzero ideal of a Dedekind domain A, then is a principal Artinian ring.
- For each, the full matrix ring over a left Artinian ring R is left Artinian.
- If R is any ring, then the polynomial ring R is not Artinian, since the ideal generated by is contained in the ideal generated by for all natural numbers n. In contrast, if R is Noetherian so is R by the Hilbert basis theorem.
- The ring of integers is a Noetherian ring but is not Artinian.
Modules over Artinian rings
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.
- is a semisimple ring, where nil is the nilradical of A.
- Every finitely generated module over A has finite length.
- A has Krull dimension zero.
- is finite and discrete.
- is discrete.
An Artinian local ring is complete. A quotient and localization of an Artinian ring is Artinian.
Simple Artinian ring
One version of the Wedderburn–Artin theorem states that a simple Artinian ring A is a matrix ring over a division ring. Indeed, let I be a minimal right ideal of A, which exists since A is Artinian. Then, since is a two-sided ideal, since A is simple. Thus, we can choose so that. Assume k is minimal with respect to that property.Now consider the map of right A-modules:
This map is surjective, since the image is a right ideal and contains 1. If it is not injective, then, say, with nonzero. Then, by the minimality of I, we have. It follows:
which contradicts the minimality of k. Hence, and thus.