Maximal ideals in a group ring when the ring is a field Is there any simple characterization of the maximal ideals in a group ring $R[G]$ when $R$ is a field, perhaps in terms of maximal subgroups of $G$?

I'm not aware of any such characterization, and at any rate, the maximal subgroups of $G$ do not determine the maximal ideals of $R[G]$ in general.
For example, if $F$ has characteristic $0$ and $G$ is finite and has exactly one maximal subgroup (say it is $(\mathbb Z_{p^k},+)$ for example) then $F[G]$ is semisimple with at least two Wedderburn components, and hence at least two maximal ideals, potentially many more maximal right and left ideals.
Perhaps you would be happy to know what is know about when $F[G]$ is local. It is known that $F[G]$ is local if $F$ has positive characteristic $p$ and $G$ is a locally-finite $p$-group, and in the other direction if $F[G]$ is local, then $F$ has characteristic $p>0$ and $G$ is a $p$-group.
From this I think it follows that there are local group rings with many distinct maximal subgroups, providing a counterpart to the example I gave in the second paragraph.
This appears as Theorem 1 in Renault's On group rings (1971). Full disclosure: I prepared the translation at this link. That's why I'm familiar with the result.
