---
tag: 0ZCAXY9563
kind: definition
name: Ascending chain condition
msc: [06A06]
defines: 0GZ6Y1H4RN
sources:
  - url: https://en.wikipedia.org/wiki/Ascending_chain_condition
    locator: Definition
---

## Definition

Let $(P, \le)$ be a partially ordered set.
The set $P$ satisfies the ascending chain condition if every sequence $a_1 \le a_2 \le a_3 \le \dots$ of elements of $P$ is eventually constant.
That is, there is an integer $N$ with $a_n = a_N$ for all $n \ge N$.

## Remarks

Equivalently, every nonempty subset of $P$ has a maximal element. This equivalence uses the axiom of dependent choice.
The set $\mathbb{N}$ with its usual order does not satisfy the condition. With the reverse order, it does.
For the set of the ideals of a ring, ordered by inclusion, the condition defines [[Noetherian rings|0MVH4Y1CRZ]].
The ideals of $\mathbb{Z}$ satisfy the condition: a chain $(a_1) \subseteq (a_2) \subseteq \dots$ means $a_2 \mid a_1$, $a_3 \mid a_2$, and so on.

## Explanations

### L1 human

The condition says that you cannot climb forever.
Every increasing chain stops growing after a finite number of steps.
For the ideals of the integers, a strictly increasing chain like $(12) \subset (6) \subset (3) \subset (1)$ must end.