TITLE: Examples of a commutative ring without an identity in which a maximal ideal is not a prime ideal
QUESTION [16 upvotes]: 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 $R^2$.

REPLY [3 votes]: I will prove the following fact.
Let $R$ be a commutative ring without an identity.
Let $M$ be a non-prime maximal ideal.
Then $R/M$ has a prime order.
Proof
Let $S = R/M$.
Since $R^2 \subset M$ by the Manny Reyes' answer, $S^2 = 0$.
Hence every subgroup of the additive group $S$ is an ideal of $S$.
Since $S$ has no non-trivial ideals, the order of $S$ must be a prime number.
QED