Maximal ideals in a group ring when the ring is a field

Question

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 $G$?

Answer 1

The maximal ideals of $R[G]$ are in canonical bijection with the isomorphism classes of simple $R[G]$-modules. In one direction, if $I$ is a maximal ideal, then $R[G]/I$ is a simple ring, hence has a unique simple module up to isomorphism, which you can lift to a simple module of $R[G]$. In the other direction, given a simple $R[G]$-module $M$, take $I$ to be its annihilator.

Whether or not this characterisation helps you in any way depends on further properties of $G$ and $R$. For example if $G$ is finite, then there is a very rich theory of simple $R[G]$-modules, equivalently of irreducible representations of $G$ over $R$, namely character theory if the characteristic of $R$ does not divide $\#G$, and the theory of Brauer characters in the other, the modular case. But in the generality that you have posed the question in, I am not sure whether anything more enlightening can be said.

Answer 2

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.