ON GENERALIZED EPI-PROJECTIVE MODULES

A module M is said to be generalized N-projective (or N-dual ojective) if, for any epimorphism g : N → X and any homomorphism f : M → X, there exist decompositions M = M₁ ⊕ M₂, N = N₁ ⊕ N₂, a homomorphism h₁ : M₁ → N₁ and an epimorphism h₂ : N₂ → M₂ such that g ◦ h₁ = f|_M1 and f ◦ h₂ = g|_N2 . This relative projectivity is very useful for the study on direct sums of lifting modules (cf. [5], [7]). In the definition, it should be noted that we may often consider the case when f to be an epimorphism. By this reason, in this paper we define relative (strongly) generalized epi-projective modules and show several results on this generalized epi-projectivity. We apply our results to the known problem when finite direct sums M₁⊕· · ·⊕M_n of lifting modules M_i (i = 1, · · · , n) is lifting.