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:


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