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)
“How have I been able to live so long outside Nature without identifying myself with it? Everything lives, moves, everything corresponds; the magnetic rays, emanating either from myself or from others, cross the limitless chain of created things unimpeded; it is a transparent network that covers the world, and its slender threads communicate themselves by degrees to the planets and stars. Captive now upon earth, I commune with the chorus of the stars who share in my joys and sorrows.”
—Gérard De Nerval (1808–1855)
“No race has the last word on culture and on civilization. You do not know what the black man is capable of; you do not know what he is thinking and therefore you do not know what the oppressed and suppressed Negro, by virtue of his condition and circumstance, may give to the world as a surprise.”
—Marcus Garvey (1887–1940)