TITLE: Examples of prime ideals that are not maximal
QUESTION [35 upvotes]: I would like to know of some examples of a prime ideal that is not maximal in some commutative ring with unity.

REPLY [4 votes]: $\Bbb Z \times \{0\} $ is a prime ideal of $\Bbb Z \times \Bbb Z$, but it is not maximal. It is contained in the proper ideal $\Bbb Z \times pZ$.