Primary Ideal - Examples and Properties

Examples and Properties

• Any prime ideal is primary, and moreover an ideal is prime if and only if it is primary and semiprime.

• Every primary ideal is primal.

• If Q is a primary ideal, then the radical of Q is necessarily a prime ideal P, and this ideal is called the associated prime ideal of Q. In this situation, Q is said to be P-primary.

• If P is a maximal prime ideal, then any ideal containing a power of P is P-primary. Not all P-primary ideals need be powers of P; for example the ideal (x, y2) is P-primary for the ideal P = (x, y) in the ring k, but is not a power of P.

• In general powers of a prime ideal P need not be P-primary. (An example is given by taking R to be the ring k/(xy − z2), with P the prime ideal (x, z). If Q = P2, then xy ∈ Q, but x is not in Q and y is not in the radical P of Q, so Q is not P-primary.) However every ideal Q with radical P is contained in a smallest P-primary ideal, consisting of all elements a such that ax is in Q for some x not in P. In particular there is a smallest P-primary ideal containing Pn, called the nth symbolic power of P.

• If A is a Noetherian ring and P a prime ideal, then the kernel of, the map from A to the localization of A at P, is the intersection of all P-primary ideals.