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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Since that my lovely knight is slain.
Wi ae lock of his yellow hair
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—Unknown. The Lament of the Border Widow (l. 25–28)
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—Flannery O’Connor (1925–1964)