Primary Ideals

Some articles on ideal, primary, primary ideal, primary ideals, ideals:

Primary Ideal
... In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n>0 ... For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number ... The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as ...
Noetherian Ring - Primary Decomposition
... In the ring of integers, an arbitrary ideal is of the form (n) for some integer n (where (n) denotes the set of all integer multiples of n) ... In this case, the ideal (n) may be written as the intersection of the ideals (piei) that is ... This is referred to as a primary decomposition of the ideal (n) ...
Lasker–Noether Theorem - Irreducible Decomposition in Rings
... The study of the decomposition of ideals in rings began as a remedy for the lack of unique factorization in number fields like , in which ... does not factor uniquely into primes, then the ideal generated by the number may still factor into the intersection of powers of prime ideals ... Failing that, an ideal may at least factor into the intersection of primary ideals ...

Famous quotes containing the words ideals and/or primary:

The real weakness of England lies, not in incomplete armaments or unfortified coasts, not in the poverty that creeps through sunless lanes, or the drunkenness that brawls in loathsome courts, but simply in the fact that her ideals are emotional and not intellectual.
Oscar Wilde (1854–1900)

The traditional American husband and father had the responsibilities—and the privileges—of playing the role of primary provider. Sharing that role is not easy. To yield exclusive access to the role is to surrender some of the potential for fulfilling the hero fantasy—a fantasy that appeals to us all. The loss is far from trivial.
Faye J. Crosby (20th century)