# 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.

### Principal ideals

Definition:Ascending Chain Condition/Principal Ideals

## Also see

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

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