\begin{definition}[Definition:Descending Chain Condition]
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 inclusion.
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 $D$.
{{expand|separate the definitions for general posets, modules, ideals, principal ideals, ...}}
\end{definition}