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### WHEN IS A QUASI-P-PROJECTIVE MODULE DISCRETE

Y. TALEBI, I. KHALILI GORJI
2008 Journal of Nonlinear Science and its Applications
It is well-known that every quasi-projective module has D 2 -condition. In this note it is shown that for a quasi-p-projective module M which is selfgenerator, duo, then M is discrete. Introduction and preliminaries Throughout, R is an associative ring with identity and right R-modules are unitary. Let M be a right R-module. A module N is called M -generated if there is an epimorphism M (I) −→ N for some index set I. In particular, N is called M -cyclic if it is isomorphism to M/L for submodule
more » ... o M/L for submodule L ⊆ M . Following [3] a module M is called self-generate if it generates all its submodules. For standard notation and terminologies, we refer to [4], [3]. Let M be a right R-module. A right R-module N is called M -p-projective if every homomorphism from N to an M -cyclic submodule of M can be lifted to an R-homomorphism from N to M . A right R-module M is called quasi-p-projective , if it is M -p-projective. A submodule A of M is said to be a small submodule of M (denoted by A M ) if for any B ⊆ M , A + B = M implies B = M . A module M is called hollow if every its submodule is small. In [2], S.Chotchaisthit showed that a quasi-p-injective module M is continouse, if M is duo and semiprefect. Here we study, when a quasi-p-projective module is discrete. Consider the following conditions for a module M which have studied in [3] : D 1 : For every submodules N of M there exist submodules K, L of M such that M = K ⊕ L and K ≤ N and N ∩ L L.