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Module Theory

Endomorphism rings and direct sum decompositions in some classes of modules

  • Alberto Facchini

Part of the Progress in Mathematics book series (PM, volume 167)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Alberto Facchini
    Pages 1-32
  3. Alberto Facchini
    Pages 33-74
  4. Alberto Facchini
    Pages 75-97
  5. Alberto Facchini
    Pages 99-111
  6. Alberto Facchini
    Pages 113-128
  7. Alberto Facchini
    Pages 129-170
  8. Alberto Facchini
    Pages 171-192
  9. Alberto Facchini
    Pages 209-235
  10. Alberto Facchini
    Pages 237-266
  11. Alberto Facchini
    Pages 267-270
  12. Back Matter
    Pages 271-288

About this book

Introduction

This expository monograph was written for three reasons. Firstly, we wanted to present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the "Krull-Schmidt Theorem" holds for ar­ tinian modules. The problem remained open for 63 years: its solution, a negative answer to Krull's question, was published only in 1995 (see [Facchini, Herbera, Levy and Vamos]). Secondly, we wanted to present the answer to a question posed by Warfield in 1975 [Warfield 75]. He proved that every finitely pre­ sented module over a serial ring is a direct sum of uniserial modules, and asked if such a decomposition was unique. In other words, Warfield asked whether the "Krull-Schmidt Theorem" holds for serial modules. The solution to this problem, a negative answer again, appeared in [Facchini 96]. Thirdly, the so­ lution to Warfield's problem shows interesting behavior, a rare phenomenon in the history of Krull-Schmidt type theorems. Essentially, the Krull-Schmidt Theorem holds for some classes of modules and not for others. When it does hold, any two indecomposable decompositions are uniquely determined up to a permutation, and when it does not hold for a class of modules, this is proved via an example. For serial modules the Krull-Schmidt Theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations. We wanted to present such a phenomenon to a wider math­ ematical audience.

Keywords

modules ring theory endomorphism ring Lattice matrices Permutation ring

Authors and affiliations

  • Alberto Facchini
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversità di UdineUdineItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-8774-8
  • Copyright Information Springer Basel AG 1998
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-0348-9769-3
  • Online ISBN 978-3-0348-8774-8
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
  • Buy this book on publisher's site
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