Concept

# Ascending chain condition

Summary
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 < \cdots of elements of P exists. Equivalently, every weakly ascending sequence :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 t
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications

Related people

Related units