Principal Ideal Domains
A principal ideal domain, or PID, is an integral domain in which every ideal is a principal ideal. A PID with only one non-zero maximal ideal is called a discrete valuation ring, or DVR, and every discrete valuation ring is a valuation ring. A valuation ring is a PID if and only if it is a DVR or a field. A value group is called discrete if and only if it is isomorphic to the additive group of the integers, and a valuation ring has a discrete valuation group if and only if it is a discrete valuation ring.
Read more about this topic: Valuation Ring
Famous quotes containing the words principal, ideal and/or domains:
“The principal office of history I take to be this: to prevent virtuous actions from being forgotten, and that evil words and deeds should fear an infamous reputation with posterity.”
—Tacitus (c. 55–c. 120)
“As to the family, I have never understood how that fits in with the other ideals—or, indeed, why it should be an ideal at all. A group of closely related persons living under one roof; it is a convenience, often a necessity, sometimes a pleasure, sometimes the reverse; but who first exalted it as admirable, an almost religious ideal?”
—Rose Macaulay (1881–1958)
“I shall be a benefactor if I conquer some realms from the night, if I report to the gazettes anything transpiring about us at that season worthy of their attention,—if I can show men that there is some beauty awake while they are asleep,—if I add to the domains of poetry.”
—Henry David Thoreau (1817–1862)