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Categories of Modules over Endomorphism Rings
Theodore G. Faticoni
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Memoirs of the American Mathematical Society
1993; 140 pp; softcover
Volume: 103
ISBN-10: 0-8218-2554-2
ISBN-13: 978-0-8218-2554-9
List Price: US$38
Individual Members: US$22.80
Institutional Members: US$30.40
Order Code: MEMO/103/492
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The goal of this work is to develop a functorial transfer of properties between a module \(A\) and the category \({\mathcal M}_{E}\) of right modules over its endomorphism ring, \(E\), that is more sensitive than the traditional starting point, \(\mathrm{Hom}(A, \cdot )\). The main result is a factorization \(\mathrm{q}_{A}\mathrm{t}_{A}\) of the left adjoint \(\mathrm{T}_{A}\) of \(\mathrm{Hom}(A, \cdot )\), where \(\mathrm{t}_{A}\) is a category equivalence and \(\mathrm{ q}_{A}\) is a forgetful functor. Applications include a characterization of the finitely generated submodules of the right \(E\)-modules \(\mathrm{Hom}(A,G)\), a connection between quasi-projective modules and flat modules, an extension of some recent work on endomorphism rings of \(\Sigma\)-quasi-projective modules, an extension of Fuller's Theorem, characterizations of several self-generating properties and injective properties, and a connection between \(\Sigma\)-self-generators and quasi-projective modules.

Readership

Research mathematicians.

Table of Contents

  • Construction of the categories
  • Tensor and \(\operatorname{Hom}\) functors
  • Category equivalences
  • Special morphisms
  • Category Equivalences for \(H_A\)
  • Projective properties in \({\mathcal M}({\mathcal P}_A)\)
  • Injective properties
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