P-injective group rings. (English) Zbl 07285982

Summary: A ring \(R\) is called right P-injective if every homomorphism from a principal right ideal of \(R\) to \(R_R\) can be extended to a homomorphism from \(R_R\) to \(R_R\). Let \(R\) be a ring and \(G\) a group. Based on a result of Nicholson and Yousif, we prove that the group ring \(\mathrm{RG}\) is right P-injective if and only if (a) \(R\) is right P-injective; (b) \(G\) is locally finite; and (c) for any finite subgroup \(H\) of \(G\) and any principal right ideal \(I\) of \(\mathrm{RH}\), if \(f\in\mathrm{Hom}_R(I_R,R_R)\), then there exists \(g\in\mathrm{Hom}_R(\mathrm{RH}_R,R_R)\) such that \(g|_I=f\). Similarly, we also obtain equivalent characterizations of \(n\)-injective group rings and F-injective group rings.


16S34 Group rings
16D50 Injective modules, self-injective associative rings
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