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:
“It is the most enduring quality, and the most ascending quality.”
—Ralph Waldo Emerson (1803–1882)
“Loyalty to petrified opinions never yet broke a chain or freed a human soul in this world—and never will.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Man is the end of nature; nothing so easily organizes itself in every part of the universe as he; no moss, no lichen is so easily born; and he takes along with him and puts out from himself the whole apparatus of society and condition extempore, as an army encamps in a desert, and where all was just now blowing sand, creates a white city in an hour, a government, a market, a place for feasting, for conversation, and for love.”
—Ralph Waldo Emerson (1803–1882)