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**Principally injective rings.**
*(English)*
Zbl 0839.16004

A ring \(R\) is called right principally injective if every \(R\)-homomorphism from a principal right ideal of \(R\) is a left multiplication by an element of \(R\).

Semiperfect, right principally injective rings with essential right socles, which are a natural generalization of the pseudo-Frobenius rings, are studied. In fact for such a ring the following interesting facts are shown: (1) that the right socle equals the left socle, and it is essential on both sides and is finitely generated on the left; (2) that the two singular ideals coincide; and (3) that such a ring has a Nakayama permutation of its basic idempotents.

Furthermore, principally injective group rings are investigated. Indeed it is proved that if the group ring \(RG\) is right principally injective, then \(R\) is right principally injective and \(G\) is locally finite; and that if \(R\) is right self-injective and \(G\) is locally finite, then the group ring \(RG\) is right principally injective.

Semiperfect, right principally injective rings with essential right socles, which are a natural generalization of the pseudo-Frobenius rings, are studied. In fact for such a ring the following interesting facts are shown: (1) that the right socle equals the left socle, and it is essential on both sides and is finitely generated on the left; (2) that the two singular ideals coincide; and (3) that such a ring has a Nakayama permutation of its basic idempotents.

Furthermore, principally injective group rings are investigated. Indeed it is proved that if the group ring \(RG\) is right principally injective, then \(R\) is right principally injective and \(G\) is locally finite; and that if \(R\) is right self-injective and \(G\) is locally finite, then the group ring \(RG\) is right principally injective.

Reviewer: J.K.Park (Pusan)

### MSC:

16D50 | Injective modules, self-injective associative rings |

16S34 | Group rings |

16L60 | Quasi-Frobenius rings |

16D25 | Ideals in associative algebras |

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |