For questions on Artinian rings, Artinian modules and related notions.
A (commutative) ring is called Artinian if every descending chain of ideals becomes stationary. For non-commutative rings, the notions of left- and right-Artinian exist, and they apply to left and right ideals respectively. An Artinian non-commutative ring is both left- and right-Noetherian.
More generally, a module is called Artinian if each descending chain of submodules becomes stationary.
A vector space is Artinian if and only if it is of finite dimension if only if is Noetherian.
