# Definition:Ascending Chain Condition

## Contents

## Definition

Let $\left({P, \leq}\right)$ be an ordered set.

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

### ACC on submodules

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

### ACC on 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.

### ACC on principal ideals

Definition:Ascending Chain Condition/Principal Ideals