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 $G$?

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.
