As known, the fact "every ideal in a unital commutative ring is contained in a maximal ideal" is proven using Zorn's lemma, but it really uses that the ring has the identity. (While using Zorn's lemma, you take a union and to show it's different from whole ring, you argue by saying that the union doesn't contain the identity)
So, I was wondering what is a counter example to this theorem in the non-unital case? (I suppose it's wrong, because I feel like you should somehow use the same argumant to prove the existance of such a maximal ideal, and if there is no identity, I don't see any reason for the union doesn't give the whole ring.) I thought about that for a while, but couldn't come up with an example yet. I would be so happy if anybody can share any such example. Thanks!
P.S. I'm sorry in advance if there is a very simple example.
