A ring is called left (respectively, right) Noetherian if it does not contain an infinite ascending chain of left (respectively, right) ideals. In this case, the ring in question is said to satisfy the ascending chain condition on left (respectively, right) ideals. A ring is said to be Noetherian if it is both left and right Noetherian. For a ring R, the following are equivalent: 1.R satisfies the ascending chain condition on ideals (i.e., is Noetherian). 2. Every ideal of R is finitely generated. 3. Every set of ideals contains a maximal element.

Artinian ring | ascending chain condition | left ideal | local ring | Noetherian module | Noether-Lasker theorem | right ideal