Artinian ring

Summary

An Artinian ring is a ring in which every decreasing chain of ideals stops after finitely many steps. For commutative rings, Artinian rings are exactly the Noetherian rings in which every prime ideal is maximal, by the Akizuki–Hopkins–Levitzki theorem. By the structure theorem for Artinian rings, a commutative Artinian ring is a finite product of Artinian local rings. Finite rings, fields and finite-dimensional algebras over a field are Artinian.