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!


1 Answer 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.

  • $\begingroup$ 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, 2014 at 13:12
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    $\begingroup$ @Geometer61 What about the direct product of two copies of a finite local ring which is not a PIR? $\endgroup$
    – user26857
    Dec 17, 2014 at 18:22
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    $\begingroup$ 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. $\endgroup$ Oct 22, 2015 at 10:29

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