A ring R is called right principally injective if every R-homomorphism
from a principal right ideal to R is left multiplication by an elemen
t of R. In this paper various properties of these rings are developed,
many extending known results. If, in addition, R is semiperfect and h
as an essential right socle, it is shown: (1) that the right socle equ
als the left socle, that this is essential on both sides and is finite
ly generated on the left; (2) that the two singular ideals coincide; a
nd (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 Bjork and Rutter. We also answer a q
uestion of Camillo about commutative principally injective rings in wh
ich every ideal contains a uniform ideal. Finally, we show that if the
group ring RG is principally injective then R is principally injectiv
e 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. (C) 1995 Academic Press, Inc.