For the ring $R=SG$, the group ring of a finite group G over an integral domain S, and a subset $I=(g-1|g \in G)$, is this subset an ideal? Is it prime? How about maximal?
2026-03-25 19:01:56.1774465316
For a group ring, finding if a subset is an ideal.
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See augmentation ideal. Obviously, augmentation map $\varepsilon$ is an epimorphism $R\to S$. Then $I=\ker(\epsilon)$ and $R/I\cong S$. So if $G$ commutative, then $I$ is a prime ideal of a commutative ring $R$. If $G$ nonabelian, then $R$ noncommutative and I don't know, what is the name for such ideals. From this clear, that $I$ maximal iff $S$ is a field (since $R/I\cong\varepsilon(R)=S$).