Definition:Ascending Chain Condition/Module
Definition
Let $R$ be a commutative ring with unity.
Let $M$ be an $R$-module.
Let $\left({D, \subseteq}\right)$ be a set of submodules of $M$ ordered by inclusion.
Then $M$ is said to have the ascending chain condition on submodules if and only if:
- Every increasing sequence $N_1 \subseteq N_2 \subseteq N_3 \subseteq \cdots$ with $N_i \in D$ eventually stabilizes: $\exists k \in \N: \forall n \in \N, n \ge k: N_n = N_{n+1}$
