# Definition:Descending Chain Condition

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## Definition

### Descending Chain Condition on Submodules

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 $D$.

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