Abstract. Let R be a ring. A right R-module M is called quasi-principally injective if it is M-principally injective.In this paper, we give some characterizations and properties of principally injective modules, which generalize results of Nicholson and Yousif. For a quasi-principally injective module M, we show: 1. For isomorphic submodules H, K of M, we have SH = SK, where S is the endomorphism ring of M. 2. M has (PC2), and consequently has (PC3). We characterize when a direct sum of P-extending modules is P-extending, and when a direct sum of a P-extending module and a semi-simple module is P-extending. We also characterize when a direct sum of FP-extending modules is FP-extending. Finally, we discuss when a direct sum of P-extending modules with relatively EC-injective is P-extending.
Keywords: Principally injective modules, extending modules.
AMS Subject classification: 16D50, 16D70, 16D80.
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