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Let $p$ be a prime number and assume a group ring $R=KG$, where $K$ is a field of characteristic $p$ and $G$ is a finite $p$-group. I want to find the set of nilpotent elements of $R$.

Thanks for any leading answer!

karparvar
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  • For some arguments, see this question. – Dietrich Burde Mar 28 '18 at 12:00
  • or this https://math.stackexchange.com/questions/1567318/prove-that-the-augmentation-ideal-in-the-group-ring-mathbbz-p-mathbbzg-is?noredirect=1&lq=1 for full answer – xsnl Mar 28 '18 at 17:57
  • @xsnl Thanks for your comment, but I want the set of all nilpotent elements (which of course contains the augmentation ideal). – karparvar Mar 28 '18 at 20:58
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    Elements outside of augmentation ideal map to nonzero elements of $K$, and (I think) that $K$ does not have any zero divisors, does it? – xsnl Mar 28 '18 at 21:09
  • @xsnl When an element $\sum n_i$ of $K$ is nonzero, what does it have to do with an element, say, $\sum n_ig$ outside the augmentation ideal being non-nilpotent? I don't understand. – karparvar Mar 28 '18 at 22:02
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    Augmentation $\epsilon$ is ring homomorphism, what is not obvious? Take any $x \in KG , \setminus ker , \epsilon $, then $\epsilon(x) \in K^*$, particularly nonzero. For any natural $n$ $\epsilon(x^n) = \epsilon(x)^n$ — still nonzero and cannot be image of zero element. – xsnl Mar 28 '18 at 23:56

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