Some Remarks on Multiplication and Projective Modules II

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Abstract

All rings are commutative with identity and all modules are unital. Let R be a ring and M an R-module. In our recent work [6] we investigated faithful multiplication modules and the properties they have in common with projective modules. In this article, we continue our study and investigate faithful multiplication and locally cyclic projective modules and give several properties for them. If M is either faithful multiplication or locally cyclic projective then M is locally either zero or isomorphic to R. We show that, if M is a faithful multiplication module or a locally cyclic projective module, then for every submodule N of M there exists a unique ideal Γ(N) ⊆ Tr(M) such that N = Γ(N)M. We use this result to show that the structure of submodules of a faithful multplication or locally cyclic projective module and their traces are closely related. We also use the trace of locally cyclic projective modules to study their endomorphisms.

Original languageEnglish
Pages (from-to)195-214
Number of pages20
JournalCommunications in Algebra
Volume41
Issue number1
DOIs
Publication statusPublished - Jan 2013

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Keywords

  • Locally cyclic projective module
  • Multiplication module
  • Ring of endormophisms
  • Trace of a module

ASJC Scopus subject areas

  • Algebra and Number Theory

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