ℵ0-Injective group rings

Abstract

Recall that a ring [Formula: see text] is called right [Formula: see text]-injective if every homomorphism from a countably generated right ideal of [Formula: see text] to [Formula: see text] can be extended to a homomorphism from [Formula: see text] to [Formula: see text]. These rings are not only a natural generalization of self-injective rings but also strongly connected with regularities of rings. Let [Formula: see text] be the group ring of a group [Formula: see text] over a ring [Formula: see text]. It is proved that [Formula: see text] is right [Formula: see text]-injective if and only if (i) [Formula: see text] is right [Formula: see text]-injective; (ii) [Formula: see text] is finite; (iii) for each countably generated right ideal [Formula: see text] of [Formula: see text], any right [Formula: see text]-homomorphism from [Formula: see text] to [Formula: see text] can be extended to a right [Formula: see text]-homomorphism from [Formula: see text] to [Formula: see text].