Definition:Ascending Chain Condition

Definition

Let $\struct {P, \le}$ be an ordered set.

Then $S$ is said to have the ascending chain condition if and only if every increasing sequence $x_1 \le x_2 \le x_3 \le \cdots$ with $x_i \in P$ eventually terminates: there is $n \in \N$ such that $x_n = x_{n 1} = \cdots$.

Submodules

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:

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

Ideals

Let $R$ be a commutative ring.

Then $R$ is said to have the ascending chain condition on ideals if and only if every increasing sequence of ideals stabilizes.

Also see

• Definition:Well-Founded Relation

• Definition:Descending Chain Condition
• Increasing Sequence in Ordered Set Terminates iff Maximal Element

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