Definition:Ascending Chain Condition/Module

Definition

Let $R$ be a commutative ring with unity.

Let $M$ be an $R$-module.

Let $\struct {D, \subseteq}$ 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

• Definition:Noetherian Module
• Definition:Descending Chain Condition/Module
• Definition:Artinian Module
• Increasing Sequence in Ordered Set Terminates iff Maximal Element

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