artinian

Artinian is a term used in abstract algebra to describe finiteness conditions, most commonly applied to rings and modules. A ring is called Artinian if it satisfies the descending chain condition on ideals: every descending sequence of ideals I1 ⊇ I2 ⊇ I3 ⊇ ... stabilizes. In the noncommutative setting, the property can be stated as left Artinian or right Artinian; a ring that is both left and right Artinian is simply Artinian. For commutative rings, the definition reduces to the same condition on all ideals.

Several consequences follow from the Artinian condition. An Artinian ring is necessarily Noetherian, so it satisfies both ascending and descending chain conditions on ideals. The Jacobson radical J(R) is nilpotent, and the quotient R/J(R) is a finite product of fields. A central structural result is the Wedderburn–Artin theorem: Artinian rings are, up to isomorphism, finite products of matrix rings over Artinian local rings (equivalently, their semisimple quotient is a finite product of simple artinian rings). These rings are therefore well-behaved from a module-theoretic perspective.

Examples include fields, and more generally any finite ring like Z/nZ. The ring k[x]/(x^m) over a field k is Artinian, while the polynomial ring k[x] is not Artinian.

Artinian also describes modules. An R-module M is Artinian if it satisfies the descending chain condition on submodules. A module has finite length precisely when it is both Artinian and Noetherian; equivalently, M has a finite composition series whose factors are simple. Finite-length modules exhibit many finiteness properties and behave predictably under submodules and quotients.