Definition:Noetherian Module

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

Let $M$ be an $A$-module.

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


 * $(1): \quad$ Every ideal $I \subset A$ is finitely generated.
 * $(2): \quad$ $A$ satisfies the ascending chain condition on subrings
 * $(3): \quad$ $A$ satisfies the maximal condition on subrings.

All of these conditions are equivalent, see Equivalence of Definitions of Noetherian Module for a proof.