Definition:Ascending Chain Condition/Module

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

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

Let $\struct {D, \subseteq}$ be a set of submodules of $M$ ordered by inclusion.

Then $M$ is said to have the ascending chain condition on submodules :


 * Every increasing sequence $N_1 \subseteq N_2 \subseteq N_3 \subseteq \cdots$ with $N_i \in D$ eventually stabilizes: $\exists k \in \N: \forall n \in \N, n \ge k: N_n = N_{n + 1}$

Also see

 * Definition:Noetherian Module
 * Definition:Descending Chain Condition/Module
 * Definition:Artinian Module
 * Increasing Sequence in Ordered Set Terminates iff Maximal Element