# Institute of Mathematics

## P-injective group rings

#### Liang Shen

###### Received April 5, 2019.   Published online April 17, 2020.

Abstract:  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 ${\rm RG}$ 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 ${\rm RH}$, if $f\in{\rm Hom}_R(I_R, R_R)$, then there exists $g\in{\rm Hom}_R({\rm RH}_R, R_R)$ such that $g|_I=f$. Similarly, we also obtain equivalent characterizations of $n$-injective group rings and F-injective group rings.
Keywords:  group ring; P-injective ring; $n$-injective ring; F-injective ring
Classification MSC:  16S34, 16D50
DOI:  10.21136/CMJ.2020.0159-19

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Affiliations:   Liang Shen, School of Mathematics, Southeast University, Jiulonghu Campus, Jiangning District, Nanjing 211189, P. R. China, e-mail: lshen@seu.edu.cn

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