Definition:Noetherian Ring

Definition
Let $A$ be a commutative ring with identity.

Then $A$ is a Noetherian ring if any of the following conditions hold:


 * $(1): \quad$ Every ideal $I \subset A$ is finitely generated.
 * $(2): \quad$ Every chain of ideals $I_1\subset I_2 \subset \dots$ stabilizes, that is, $\exists N$ such that $\forall n \ge N$, $I_n = I_{n+1}$.
 * $(3): \quad$ Every set of ideals has a maximal element.

All of these conditions are equivalent, and to state that a ring is Noetherian means all of these conditions are true.

See Equivalence of Definitions of Noetherian Ring for a proof.