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 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 RH, if f ∈ Hom R ( I R , R R ), then there exists g ∈ Hom R (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.

中文翻译：

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P-注入群环

如果从 R 的主右理想到 RR 的每个同态都可以扩展为从 RR 到 RR 的同态，则环 R 被称为右 P 射。设 R 为环，G 为群。基于 Nicholson 和 Yousif 的结果，我们证明群环 RG 是右 P 射的当且仅当 (a) R 是右 P 射；(b) G 是局部有限的；(c) 对于 G 的任意有限子群 H 和 RH 的任意主右理想 I，如果 f ∈ Hom R ( IR , RR )，则存在 g ∈ Hom R (RH R , RR ) 使得 g I = f . 类似地，我们也得到了n-射入群环和F-射入群环的等价表征。