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)
“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)
“...Negroes must concern themselves with every single means of struggle: legal, illegal, passive, active, violent and non- violent.... They must harass, debate, petition, boycott, sing hymns, pray on steps—and shoot from their windows when the racists come cruising through their communities.... The acceptance of our condition is the only form of extremism which discredits us before our children [ellipses in source].”
—Lorraine Hansberry (1930–1965)