ℵ0-Injective group rings

Recall that a ring R is called right aleph(0)-injective if every homomorphism from a countably generated right ideal of R to R-R can be extended to a homomorphism from R-R to R-R. These rings are not only a natural generalization of self-injective rings but also strongly connected with regularities of rings. Let RG be the group ring of a group G over a ring R. It is proved that RG is right aleph(0)-injective if and only if (i) R is right aleph(0)-injective; (ii) C is finite; (iii) for each countably generated right ideal I of RC, any right R-homomorphism from I-R to R-R can be extended to a right R-homomorphism from RG(R) to R-R.