Is there a maximal ideal that isn't prime if the ring is non commutative?

Question

In the proof of "every maximal ideal is prime" we use commutativity at some point. So I was wondering if there were examples of non abelian rings with a maximal (one-sided) ideal that isn't prime...

Answer 1

As stated in the comments, $M_2(F)$, where $F$ is a field, the zero ideal is a maximal ideal which does not satisfy the commutative definition of a prime ideal.

However, that is kind of a cheat because prime ideals are defined differently for noncommutative rings: an ideal $P$ is called prime if for any two ideals $A,B$, $AB\subseteq P$ implies $A\subseteq P$ or $B\subseteq P$.

So the "right" version of this question asks if maximal ideals are prime with respect to this definition. The general definition reduces to the "commutative definition" in commutative rings, and using this definition, maximal ideals are prime ideals in noncommutative rings.

So while I guess you did not know what waters you were wading into when you asked the question, I think this latter explanation is the one you should prefer.

You also mentioned the possibility of prime one-sided ideals. Then again you'll have to pick which definition you want to discuss, for there are many variants on what constitutes a one-sided prime ideal.