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:
“Life seems to be an experience in ascending and descending. You think youre beginning to live for a single aimfor self-development, or the discovery of cosmic truthswhen all youre really doing is to move from place to place as if devoted primarily to real estate.”
—Margaret Anderson (18861973)
“To avoid tripping on the chain of the past, you have to pick it up and wind it about you.”
—Mason Cooley (b. 1927)
“Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to do it. The artist works out his own formulas; the interest of science lies in the art of making science.”
—Paul Valéry (18711945)