What are primary 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 an intersection of finitely ...

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 ... If a number 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 ...

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) ...