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!


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.

  • $\begingroup$ This is indeed an example of a finite local ring that is not a PIR. Do you know also an example of a non-local finite ring that is not a PIR? $\endgroup$ – Geometer61 Dec 17 '14 at 13:12
  • $\begingroup$ @Geometer61 What about the direct product of two copies of a finite local ring which is not a PIR? $\endgroup$ – user26857 Dec 17 '14 at 18:22
  • $\begingroup$ This is such a neat example. Has any work been done on $\mathbb{F}_2[X]/I$ where $X$ is a set and $I$ is the ideal of $\mathbb{F}_2[X]$ generated by $\{ab \mid a,b \in X\}$? Do these rings have an accepted named? Anyway, very cool. $\endgroup$ – goblin Oct 22 '15 at 10:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.