Whenever I say ideal in this question I'm talking about two sided-ideals.
Does there exist a concrete example of a non-commutative ring with $1$ without maximal ideals?
We know that if $R$ is a commutative ring with a multiplicative identity element then it does have maximal ideals. We know that if we drop the assumption that $R$ is unital, then there are commutative rings without maximal ideals. But what if we keep the assumption that $R$ has a multiplicative identity element but drop the assumption that it's commutative? Any interesting counter-examples?