I would like to have an example (or a class of examples) of a finite ring (with unit, not necessarily commutative) that is not a principal ideal ring (if possible an example of a local ring and of a nonlocal ring).
I know that finite chain rings are local principal ideal rings. So the example I'm looking for cannot be a chain ring (but may be local or not). I'm totally not familiar with ring theory but an answer to this question can help me a lot in a problem of finite geometry. So any help is welcome!