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:
“It is the most enduring quality, and the most ascending quality.”
—Ralph Waldo Emerson (1803–1882)
“The name of the town isn’t important. It’s the one that’s just twenty-eight minutes from the big city. Twenty-three if you catch the morning express. It’s on a river and it’s got houses and stores and churches. And a main street. Nothing fancy like Broadway or Market, just plain Broadway. Drug, dry good, shoes. Those horrible little chain stores that breed like rabbits.”
—Joseph L. Mankiewicz (1909–1993)
“Virginity is now a mere preamble or waiting room to be got out of as soon as possible; it is without significance. Old age is similarly a waiting room, where you go after life’s over and wait for cancer or a stroke. The years before and after the menstrual years are vestigial: the only meaningful condition left to women is that of fruitfulness.”
—Ursula K. Le Guin (b. 1929)