A simple algebra that is not semisimple

Question

I was said that can exists simple algebras $A$ that are not semi-simples in the following sense:

1. $A$ is simple, i.e. doesn't have non trivial ideals;
2. $A$ is not semi-simple, i.e. is not a semi-simple module over itself.

Does anybody has an example? Thanks in advance

Answer 1

Yes, the Weyl algebra $A = \mathbb{C}[x, \partial]/(\partial x - x \partial - 1)$ is a standard example. Proving that it's simple but not semisimple is a nice exercise.