Definition:Ascending Chain Condition/Module

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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: $\exists k \in \N: \forall n \in \N, n \ge k: N_n = N_{n+1}$


Also see