Rings, Intersection and Prime Ideals

Intersection and Prime Ideals

If a prime ideal P is the intersection of finitely many ideals, P equals one of the intersecting ideals.

Since the intersection contains the product, P contains the product, and since P is prime it contains one of the ideals.  yet all of the ideals contain P, so P is one of the ideals.

The intersection of prime ideals need not be prime, as shown by 2Z∩3Z = 6Z in the integers.

Descending Chains of Prime Ideals

The intersection of a descending chain of prime ideals is prime.  If xRy is in every ideal in the chain, each ideal contains x or y.  If x is no longer present, after a while, y is present throughout the rest of the chain.  Thus the intersection is prime.

By zorn's lemma, every prime ideal contains a minimal prime ideal.  Keep taking smaller prime ideals, or take the intersection of descending chains of prime ideals, until you reach a minimal prime ideal.

In a domain, the minimal prime ideal is always zero.  If xxy = 0 then either x or y is 0, and 0 becomes a prime ideal.