Ascending chain condition


In mathematics, the ascending chain condition and descending chain condition 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 P is said to satisfy the ascending chain condition if no strictly ascending sequence
of elements of P exists.
Equivalently, given any weakly ascending sequence
of elements of P there exists a positive integer n such that
Similarly, P is said to satisfy the descending chain condition if there is no infinite descending chain of elements of P. Equivalently, every weakly descending sequence
of elements of P eventually stabilizes.

Comments

Consider the ring
of integers. Each ideal of consists of all multiples of some number. For example, the ideal
consists of all multiples of. Let
be the ideal consisting of all multiples of. The ideal is contained inside the ideal, since every multiple of is also a multiple of. In turn, the ideal is contained in the ideal, since every multiple of is a multiple of. However, at this point there is no larger ideal; we have "topped out" at.
In general, if are ideals of such that is contained in, is contained in, and so on, then there is some for which all. That is, after some point all the ideals are equal to each other. Therefore the ideals of satisfy the ascending chain condition, where ideals are ordered by set inclusion. Hence is Noetherian ring.