Noetherian Ring
For (possibly non-commutative) ring
, we say is left noetherian if every left ideal of is finitely generated, and right noetherian if every right ideal of is finitely generated. If is both left and right noetherian, we say is noetherian.
Equivalently, we can formulate the definition in terms of the ascending chain condition (ACC). That is,is left (right) noetherian if every ascending chain of left (right) ideals stabilizes eventually.
Remark
For commutative rings, the notions of left and right noetherian coincide.