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.