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)
“The years seemed to stretch before her like the land: spring, summer, autumn, winter, spring; always the same patient fields, the patient little trees, the patient lives; always the same yearning; the same pulling at the chain—until the instinct to live had torn itself and bled and weakened for the last time, until the chain secured a dead woman, who might cautiously be released.”
—Willa Cather (1873–1947)
“Humility is a grace that shines in a high condition but cannot, equally, in a low one because a person in the latter is already, perhaps, too much humbled.”
—Samuel Richardson (1689–1761)