On small injective group rings

Abstract

A ring [Formula: see text] is called right small injective if every homomorphism from a small right ideal of [Formula: see text] to [Formula: see text] can be extended to a homomorphism from [Formula: see text] to [Formula: see text]. Recall that a right ideal [Formula: see text] of [Formula: see text] is called small if for any proper right ideal [Formula: see text] of [Formula: see text], [Formula: see text]. It is proved that if [Formula: see text] is a locally finite group and the group ring [Formula: see text] is right small injective, then [Formula: see text] must be right small injective. The converse is not true even if [Formula: see text] is a finite group. An example is given to show a right small injective group ring [Formula: see text] whose coefficient ring [Formula: see text] is not right small injective. In the end, let [Formula: see text] be a finite group, an equivalent characterization is given for [Formula: see text] to be right small injective.