Ascending Chain Condition

Ascending Chain Condition

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.

Read more about Ascending Chain Condition:  Definition

Famous quotes containing the words ascending, chain and/or condition:

    To do good is like ascending a mountain; to do evil as easy as following an avalanche.
    Chinese proverb.

    Oh yes, that’s right. They chain up wild animals. That’s all I am, an animal.
    John Elder [Anthony Hinds], British screenwriter, and Terence Fisher. Leon (Oliver Reed)

    Man does not understand nor accept immortality except on condition of self-remembrance.
    Victor Hugo (1802–1885)