NONNIL-P-COHERENT RINGS AND NONNIL-PP-RINGS

. Let R be a commutative ring with nonzero identity. An Rmodule M is said to be Phi-P-flat if, for any s E R \ Nil(R) and any x E M such that x = 0, we have x E (0 : s)M. An R-module M is said to be nonnil-P-injective if, for any a E R \ Nil(R), every homomorphism from Ra to M extends to a homomorphism from R to M. Then R is said to be a nonnil-P-coherent ring (resp., Phi-PF, nonnil-PP-ring) if, for any a E R \ Nil(R), Ra is a finitely presented (resp., flat, projective) module. In this paper, we study nonnil-P-coherent rings, Phi-PF-rings, and nonnilPP-rings using Phi-P-flat and nonnil-P-injective modules.