Principally Injective Rings

Abstract

A ring R is called right principally injective if every R-homomorphism from a principal right ideal to R is left multiplication by an element of R. In this paper various properties of these rings are developed, many extending known results. If, in addition, R is semiperfect and has an essential right socle, it is shown: (1) that the right socle equals the left socle, that this is essential on both sides and is finitely generated on the left; (2) that the two singular ideals coincide; and (3) that R admits a Nakayama permutation of its basic idempotents. These rings are a natural generalization of the pseudo-Frobenius rings, and our work extends results of Björk and Rutter. We also answer a question of Camillo about commutative principally injective rings in which every ideal contains a uniform ideal. Finally, we show that if the group ring RG is principally injective then R is principally injective and G is locally finite; and that if R is right selfinjective and G is locally finite then RG is principally injective, extending results of Farkas.