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

Primary Ideal

... In mathematics, specifically commutative algebra, a proper

... 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

... 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

... 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)

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