This question already has an answer here:
- A maximal ideal is always a prime ideal? 3 answers
The question formuled in the exam was exactly:
''Every maximal ideal is a prime ideal''
Maximal and prime ideals are defined for a commutative ring R, but the proof I have for maximal ideal $\Rightarrow$ prime ideal needs that R is a commutative and unitary ring, because it uses that $R/I$ is field iff $I$ is maximal, and that only happens if R is both commutative and unitary.
Then my question is:
Is there a proof for maximal ideal $\Rightarrow$ prime ideal for R a commutative ring (not neccesarily unitary)?
If not, there must be a counterexample with a maximal ideal which isn't prime in a R commutative (not unitary) ring, right?
I know there are more posts like this, but the difference is that those posts suppose R as a commutative and unitary ring (that's the difference with this one, just to be clear).