This question was inspired by Qiaochu's recent question, Which commutative groups are the group of units of some field? - my question is close to being the inverse of it.

As mentioned here, given a ring $R$, the functor $GrpRing:Grp\rightarrow R$-$Alg$ taking a group $G$ to the group ring $R[G]$ is left adjoint to the functor $GrpUnits:R$-$Alg\rightarrow Grp$ taking an $R$-algebra to its group of units. What is the essential image of $GrpRing$, i.e., which $R$-algebras are isomorphic to the group ring of some group over $R$?

One might ask more generally, when is a ring $R$ a group ring over some ring, not fixed at the outset? (Obviously, any ring $R$ is isomorphic to $R[1]$, the group ring of the trivial group over itself, but let's exclude this trivial case.)