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 every strictly ascending sequence of elements eventually terminates.[clarification needed] Equivalently, given any weakly ascending sequence

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

there exists a positive integer n such that

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

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

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

of elements of P eventually stabilizes.