In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.
Read more about Structure Theorem For Finitely Generated Modules Over A Principal Ideal Domain: Statement, Proofs, Corollaries, Uniqueness
Famous quotes containing the words structure, theorem, generated, principal, ideal and/or domain:
“If rightly made, a boat would be a sort of amphibious animal, a creature of two elements, related by one half its structure to some swift and shapely fish, and by the other to some strong-winged and graceful bird.”
—Henry David Thoreau (1817–1862)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“Here [in London, history] ... seemed the very fabric of things, as if the city were a single growth of stone and brick, uncounted strata of message and meaning, age upon age, generated over the centuries to the dictates of some now all-but-unreadable DNA of commerce and empire.”
—William Gibson (b. 1948)
“This place is the Devil, or at least his principal residence, they call it the University, but any other appellation would have suited it much better, for study is the last pursuit of the society; the Master eats, drinks, and sleeps, the Fellows drink, dispute and pun, the employments of the undergraduates you will probably conjecture without my description.”
—George Gordon Noel Byron (1788–1824)
“If we love-and-serve an ideal we reach backward in time to its inception and forward to its consummation. To grow is sometimes to hurt; but who would return to smallness?”
—Sarah Patton Boyle, U.S. civil rights activist and author. The Desegregated Heart, part 3, ch. 3 (1962)
“Without metaphor the handling of general concepts such as culture and civilization becomes impossible, and that of disease and disorder is the obvious one for the case in point. Is not crisis itself a concept we owe to Hippocrates? In the social and cultural domain no metaphor is more apt than the pathological one.”
—Johan Huizinga (1872–1945)