Principally quasi-injective modules

An R-module M is called principally quasi-injective if each R-homomorphism from a principal submodule of M to M can be extended to an endomorphism of M. Many properties of principally injective rings and quasi-injective modul es are extended to these modules. As one application, we show that, for a f inite-dimensional quasi-injective module M in which every maximal uniform s ubmodule is fully invariant, there is a bijection between the set of indeco mposable summands of M and the maximal left ideals of the endomorphism ring of M.