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
Full Text:

References:

 [1] Connell, I. G., On the group ring, Can. J. Math. 15 (1963), 650-685 [2] Farkas, D. R., A note on locally finite group algebras, Proc. Am. Math. Soc. 48 (1975), 26-28 [3] Ikeda, M., Some generalizations of quasi-Frobenius rings, Osaka Math. J. 3 (1951), 227-239 [4] Koşan, M. T.; Lee, T.-K.; Zhou, Y., On modules over group rings, Algebr. Represent. Theory 17 (2014), 87-102 [5] Nicholson, W. K.; Yousif, M. F., Principally injective rings, J. Algebra 174 (1995), 77-93 [6] Nicholson, W. K.; Yousif, M. F., Quasi-Frobenius Rings, Cambridge Tracts in Mathematics 158, Cambridge University Press, Cambridge (2003) [7] Renault, G., Sur les anneaux des groupes, C. R. Acad. Sci. Paris, Sér. A 273 (1971), 84-87 French
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.