In a commutative ring with an identity, every maximal ideal is a prime ideal. However, if a commutative ring does not have an identity, I'm not sure this is true. I would like to know the counterexamples, if any. The more examples, the better.

**EDIT**
I would like to know the counterexamples other than $2\mathbb{Z}$.
The more examples, the better.

**EDIT**
I also would like to know the counterexamples that are not given in the Arturo Magidin's answer if any, namely an example of a non-prime maximal ideal which does not contain $R^2$.