On Almost Projective Modules

Nil Orhan Ertaş

doi:10.3390/axioms10010021

Abstract

In this note, we investigate the relationship between almost projective modules and generalized projective modules. These concepts are useful for the study on the finite direct sum of lifting modules. It is proved that; if M is generalized N-projective for any modules M and N, then M is almost N-projective. We also show that if M is almost N-projective and N is lifting, then M is im-small N-projective. We also discuss the question of when the finite direct sum of lifting modules is again lifting.

Highlights

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Kuratomi gave equivalent conditions for a module with exchange decomposition M = ⊕in=1 Mi to be lifting in terms of the relatively generalized projectivity of the direct summand of M in [5]
We give the relation between almost projective modules and some kind of generalized projective modules. We apply these results to a question when the finite direct sum of lifting module is lifting

Summary

Preliminaries and Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Injectivity, and other related concepts have been studied extensively in recent years by many authors, especially by Harada and his collaborators. M is called a lifting module if, for every submodule N of M, there exists a direct summand. Direct sums of lifting modules are not lifting. The generalized projectivity has roots in the study of direct sums of lifting modules. Kuratomi gave equivalent conditions for a module with exchange decomposition M = ⊕in=1 Mi to be lifting in terms of the relatively generalized projectivity of the direct summand of M in [5]. Kuratomi proved that finite direct sums of lifting modules are again lifting, when the distinct pairs of decomposition are relatively projective. We showed that generalized projectivity implies almost projectivity

Result

Almost Projectivity

Generalized Small Projective Modules