A note on injective group rings

In this note we shall prove the following result: Let K be a commutative ring (with identity) and torsion free as a Z-module.Then if G is any group then the group algebra K(G) is left self-infective if and only if K is self-infective and G is finite.Part "if" was first proved by Eilenberg-Nakayama (see [2]).for a commutative ring K, then K is self-injective and G is a locally finite group.So, the main result to be proved here is that if A is a torsion free commutative ring and K(G) is left selfinjective then G is a finite group.In the present proof we shall need the following result from homological algebra (see [l, Chapter VI, Exercise 10]):If 0: T-*S is a ring morphism and if 5 is (via 0) a flat F-module then any left 5-module which is 5-injective is F-injective.Next we prove several partial results.