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:
“The true thrift is always to spend on the higher plane; to invest and invest, with keener avarice, that he may spend in spiritual creation, and not in augmenting animal existence. Nor is the man enriched, in repeating the old experiments of animal sensation; nor unless through new powers and ascending pleasures he knows himself by the actual experience of higher good to be already on the way to the highest.”
—Ralph Waldo Emerson (1803–1882)
“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)
“Our condition as men is risky and ticklish enough. One can not be sure of himself and his fortune an hour, but he may be whisked off into some pitiable or ridiculous plight.”
—Ralph Waldo Emerson (1803–1882)