Given $G$ a finite group and $K$ a field of characteristic zero such that all idempotents in $KG$ are central, is it true that $KG$ has no nilpotent element or equivalently $KG$ has only division ring in its Wedderburn decomposition?
2026-03-27 21:19:07.1774646347
All idempotents are central then $KG$ has no nilpotent
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Yes. If there is a factor $M_r(D)$ with $r\geq2$ in the decomposition, then any elementary matrix $E_{ii}$ is a non central idempotent.