Is every prime ideal of $\displaystyle Z \oplus Z$ also a maximal ideal?

If so, how can this be justified?

If not, can anyone think of example?

Obviously, every maximal ideal is also prime. Also, if we were in an finite integral domain, then every prime ideal would be maximal... but $\displaystyle Z \oplus Z$ has zero divisors, so we are not in an integral domain, and it is also infinite.

Any help much appreciated!

Thanks.