ON GENERALIZED EPI-PROJECTIVE MODULES

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Abstract

<p>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 = M<sub>1</sub> ⊕ M<sub>2</sub>, N = N<sub>1</sub> ⊕ N<sub>2</sub>, a homomorphism h<sub>1</sub> : M<sub>1</sub> → N<sub>1</sub> and an epimorphism h<sub>2</sub> : N<sub>2</sub> → M<sub>2</sub> such that g ◦ h<sub>1</sub> = f|<sub>M<sub>1</sub></sub> and f ◦ h<sub>2</sub> = g|<sub>N<sub>2</sub></sub> . 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 M<sub>1</sub>⊕· · ·⊕M<sub>n</sub> of lifting modules M<sub>i</sub> (i = 1, · · · , n) is lifting.</p>

Journal

  • Mathematical Journal of Okayama University

    Mathematical Journal of Okayama University 52(1), 111-122, 2010-01

    Department of Mathematics, Faculty of Science, Okayama University

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