Artinring

Artinring, more commonly called an Artinian ring, is a ring in which the descending chain condition on two‑sided ideals holds: every descending sequence of two‑sided ideals stabilizes. Equivalently, the ring is Artinian as a module over itself. In particular, every left (and hence every right) ideal is finitely generated, and by the Hopkins–Levitzki theorem Artinian rings are Noetherian on both sides.

In the commutative case, Artinring is equivalent to having Krull dimension zero and to being a finite product of Artin local rings. For a commutative Artin ring R, the Jacobson radical J(R) is nilpotent and the quotient R/J(R) is a finite product of fields. Consequently, every commutative Artin ring has finite length as a module over itself and is a finite-dimensional algebra over its residue fields.

Structure and decomposition generalize similarly in the noncommutative setting: Artin rings have a semisimple quotient R/J(R), and R has finite length as a module over itself. In the commutative case this often translates into a finite product of local components, each with a nilpotent maximal ideal.

Examples include Z/nZ for any positive integer n, finite direct products such as Z/4Z × Z/9Z, and local examples like k[x]/(x^n) for a field k. Rings of polynomials in infinitely many variables over a field or rings of infinite length are not Artinrings.

Artinring theory plays a role in the study of zero-dimensional schemes, representation theory of finite-dimensional algebras, and the structural analysis of more general rings through their Artinian components.