Group of units in a finite ring

Let G be any group with n elements, where n is a power of a prime or any product of prime powers, not divisible by 4. In this paper we find all nonisomorphic rings with its group of units isomorphic to G and also find all groups G with n elements which can be groups of units of a finite ring. We say that a group G is indecomposable, if we cannot write G=HK for some proper, nontrivial subgroups H and K. We find all finite rings with indecomposable, solvable group of units and find all finite rings with G=1+J, where J is the Jacobson radical of R. These results are obtained through a study of p-rings and idempotents in rings yielding decompositions of rings and decompositions of groups of units of rings into product of subgroups.