The ascending chain condition
(ACC) and descending chain condition
(DCC) are finiteness properties satisfied by certain algebraic structures, most importantly, ideals
in a commutative ring
. 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.
A partially ordered set
is said to satisfy the ascending chain condition
(ACC) if every ascending chain a1
≤ ... of elements of P
is eventually stationary, that is, there is some positive integer n
such that am
for all m
. Similarly, P
is said to satisfy the descending chain condition
(DCC) if every descending chain a1
≥ ... of elements of P
is eventually stationary (that is, there is no infinite descending chain
- The descending chain condition on P is equivalent to P being well-founded: every nonempty subset of P has a minimal element (also called the minimal condition).
- Similarly, the ascending chain condition is equivalent to P being converse well-founded: every nonempty subset of P has a maximal element (the maximal condition).
- Every finite poset satisfies both ACC and DCC.
- A totally ordered set that satisfies the descending chain condition is called a well-ordered set.