Definition:Descending Chain Condition/Module

Definition

Let $A$ be a commutative ring with unity.

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

Let $\struct {D, \supseteq}$ be a set of submodules of $M$ ordered by the subset relation.

Then the hypothesis:

Every decreasing sequence $N_1 \supseteq N_2 \supseteq N_3 \supseteq \cdots$ with $N_i \in D$ eventually terminates: there is $k \in \N$ such that $N_k = N_{k + 1} = \cdots$

is called the descending chain condition on the submodules in $D$.

Also see

• Definition:Ascending Chain Condition/Module

• Definition:Well-Founded Relation