Then there are three kinds of prime ideals in A:

The nilradical is equal to the intersection of all the ring's prime ideals.

The sum of two prime ideals is not necessarily prime.

It turns out that the irreducible varieties correspond to prime ideals.

By taking all prime ideals, one thus gets the whole collection of ordinary and generic points.

This makes the study of the prime ideals in O particularly important.

Thus the only prime ideals are the primes in ℤ.

The set of all prime ideals of a ring, ordered by inclusion.

Also consider the prime ideals generated by the non-invertible factors.

Any ideal is the product of prime ideals, and in one way only: