Module Theory

Endomorphism rings and direct sum decompositions in some classes of modules

  • Alberto Facchini

Part of the Modern Birkhäuser Classics book series (MBC)

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 271-285

About this book


The purpose of this expository monograph is three-fold. First, the solution of a problem posed by Wolfgang Krull in 1932 is presented. He asked whether what is now called the "Krull-Schmidt Theorem" holds for artinian modules. A negative answer was published only in 1995 by Facchini, Herbera, Levy and Vámos. Second, the answer to a question posed by Warfield in 1975, namely, whether the Krull-Schmidt-Theorem holds for serial modules, is described. Facchini published a negative answer in 1996. The solution to the Warfield problem shows an interesting behavior; in fact, it is a phenomena so rare in the history of Krull-Schmidt type theorems that its presentation to a wider mathematical audience provides the third incentive for this monograph. Briefly, the Krull-Schmidt-Theorem holds for some, not all, classes of modules. When it does hold, any two indecomposable decompositions are uniquely determined up to one permutation. For serial modules the theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations.

Apart from these issues, the book addresses various topics in module theory and ring theory, some now considered classical (such as Goldie dimension, semiperfect rings, Krull dimension, rings of quotients, and their applications) and others more specialized (such as dual Goldie dimension, semilocal endomorphism rings, serial rings and modules, exchange property, (sigma)-pure-injective modules). Open problems conclude the work.


Besides its research value, this book is a considerable addition to the list of fundamental books in rings and modules. It is written carefully with necessary backgrounds developed in a logical way. There are many fundamental points worked out in the book. All these make the book an excellent contribution to the development of module and ring theory, and a source of reference. Algebraists will certainly enjoy themselves in reading this book.

(Zentralblatt MATH)

This excellent book on module and ring theory (…) contains three kind of topics: classical topics, (…) specialized topics, (…) the solutions to two famous problems, together with the presentation of a rare phenomenon.



Artinian Modules Krull-Schmidt-Theorem Module Theory Permutation endomorphism ring ring theory

Authors and affiliations

  • Alberto Facchini
    • 1
  1. 1., Department of MathematicsUniversity of PadovaPadovaItaly

Bibliographic information