This question was inspired by Qiaochu's recent question, [Which commutative groups are the group of units of some field?][2] - my question is close to being the inverse of it. 

As mentioned [here][3], 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.)


  [2]: http://mathoverflow.net/questions/13017/which-commutative-groups-are-the-group-of-units-of-some-field
  [3]: http://en.wikipedia.org/wiki/Group_ring#Category_theory