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.
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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)
“Man ... cannot learn to forget, but hangs on the past: however far or fast he runs, that chain runs with him.”
—Friedrich Nietzsche (1844–1900)
“The condition every art requires is, not so much freedom from restriction, as freedom from adulteration and from the intrusion of foreign matter.”
—Willa Cather (1876–1947)