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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.
MSC:
 16S34 Group rings 16D50 Injective modules, self-injective associative rings
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References:
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