Example of a ring with a unique two sided maximal ideal which is not a local ring (that is it has more than one left or right maximal ideals).

Question

Let $R$ be a ring (possibly non-commutative).

Definition $R$ is called a local ring if it has a unique left(and equivalently right) maximal ideal.

I am looking for an example of a ring (obviously non-commutative) which has a unique two-sided maximal ideal but is not local.

Answer 1

The ring of $2×2$ matrices over reals is simple so it has one maximal two sided ideal, 0, but it has 2 maximal left (right) ideals.

Answer 2

Literally every simple ring that isn’t a division ring, so for example every matrix ring over a division ring ($n>1$ of course), and the first Weyl algebra.

Beyond those also, every ring of linear transformations of an infinite dimensional vector space has a unique maximal two sided ideal, but isn’t local (and also isn’t simple). In fact their two sided ideals are linearly ordered.

And if you have any maximal ideal $M$ of a ring $R$ that you know isn’t maximal as a right ideal, and it isn’t nilpotent, then you can force it to be a unique maximal ideal in the quotient ring $R/M^n$, which won’t be local because of right ideal correspondence.