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 you’re beginning to live for a single aim—for self-development, or the discovery of cosmic truths—when all you’re really doing is to move from place to place as if devoted primarily to real estate.”
—Margaret Anderson (1886–1973)
“You know, he wanted to shoot the Royal Family, abolish marriage, and put everybody who’d been to public school in a chain gang. Yeah, he was a idealist, your dad was.”
—David Mercer, British screenwriter, and Karel Reisz. Mrs. Dell (Irene Handl)
“... the condition of women affords, in all countries, the best criterion by which to judge the character of men.”
—Frances Wright (1795–1852)