Ascending chain condition

In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin.

The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.

Definition

A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence

a

1

<

a

2

<

a

3

<

⋯

{\displaystyle a_{1}<a_{2}<a_{3}<\cdots }

of elements of P exists.

Equivalently, every weakly ascending sequence

a

1

≤

a

2

≤

a

3

≤

⋯

,

{\displaystyle a_{1}\leq a_{2}\leq a_{3}\leq \cdots ,}

of elements of P eventually stabilizes, meaning that there exists a positive integer n such that

a

n

=

a

n

+

1

=

a

n

+

2

=

⋯

.

{\displaystyle a_{n}=a_{n+1}=a_{n+2}=\cdots .}

Similarly, P is said to satisfy the descending chain condition (DCC) if there is no infinite strictly descending chain of elements of P. Equivalently, every weakly descending sequence

a

1

≥

a

2

≥

a

3

≥

⋯

{\displaystyle a_{1}\geq a_{2}\geq a_{3}\geq \cdots }

of elements of P eventually stabilizes.

Comments

Assuming the axiom of dependent choice, the descending chain condition on a (possibly infinite) poset P is equivalent to P being well-founded: every nonempty subset of P has a minimal element (also called the minimal condition or minimum condition). A totally ordered set that is well-founded is a well-ordered set.

Similarly, the ascending chain condition is equivalent to P being converse well-founded (again, assuming dependent choice): every nonempty subset of P has a maximal element (the maximal condition or maximum condition).