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:
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—Margaret Anderson (1886–1973)
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—Joseph L. Mankiewicz (1909–1993)
“I have heard that whoever loves is in no condition old. I have heard that whenever the name of man is spoken, the doctrine of immortality is announced; it cleaves to his constitution. The mode of it baffles our wit, and no whisper comes to us from the other side. But the inference from the working of intellect, hiving knowledge, hiving skill,—at the end of life just ready to be born,—affirms the inspirations of affection and of the moral sentiment.”
—Ralph Waldo Emerson (1803–1882)