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:
“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)
“Man ... cannot learn to forget, but hangs on the past: however far or fast he runs, that chain runs with him.”
—Friedrich Nietzsche (1844–1900)
“Humility is a grace that shines in a high condition but cannot, equally, in a low one because a person in the latter is already, perhaps, too much humbled.”
—Samuel Richardson (1689–1761)