Definition:Descending Chain Condition
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Definition
Let $A$ be a commutative ring with unity.
Let $M$ be an $A$-module.
Let $(D,\supseteq)$ be a set of submodules of $M$ ordered by inclusion.
Then the hypothesis
- Every increasing 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 $D$.
