Definition:Artinian Module

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

Let $M$ be an $A$-module

Then $A$ is a Artinian module if either of the following conditions hold:


 * $(1): \quad$ $A$ satisfies the descending chain condition on subrings
 * $(2): \quad$ $A$ satisfies the minimal condition on subrings.

Also see
These conditions are equivalent.

See Equivalence of Definitions of Artinian Module for a proof.