Example of finite ring with a non principal ideal

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!

Solution 1:

An easy example would be the ring $$\mathbb{F}_2[x,y]/(x^2,xy,y^2)=\mathbb{F}_2[\tilde{x},\tilde{y}]=\{0,\,1,\,\tilde{x},\,\tilde{y},\,1+\tilde{x},\,1+\tilde{y},\,\tilde{x}+\tilde{y},\,1+\tilde{x}+\tilde{y}\}$$ where the ideal $$I=(\tilde{x},\tilde{y})=\{0,\tilde{x},\tilde{y},\tilde{x}+\tilde{y}\}$$ cannot be generated by a single element.