© 2004

Rings, Modules, and the Total

  • Accessible to anyone with a basic knowledge of ring and module theory

  • A short introduction to torsion-free Abelian groups is included


Part of the Frontiers in Mathematics book series (FM)

Table of contents

  1. Front Matter
  2. Friedrich Kasch, Adolf Mader
    Pages 1-7
  3. Friedrich Kasch, Adolf Mader
    Pages 9-38
  4. Friedrich Kasch, Adolf Mader
    Pages 39-68
  5. Friedrich Kasch, Adolf Mader
    Pages 69-100
  6. Friedrich Kasch, Adolf Mader
    Pages 101-129
  7. Back Matter

About this book


In a nutshell, the book deals with direct decompositions of modules and associated concepts. The central notion of "partially invertible homomorphisms”, namely those that are factors of a non-zero idempotent, is introduced in a very accessible fashion. Units and regular elements are partially invertible. The "total” consists of all elements that are not partially invertible. The total contains the radical and the singular and cosingular submodules, but while the total is closed under right and left multiplication, it may not be closed under addition. Cases are discussed where the total is additively closed. The total is particularly suited to deal with the endomorphism ring of the direct sum of modules that all have local endomorphism rings and is applied in this case. Further applications are given for torsion-free Abelian groups.


Algebra Group Theory Modules Rings Abelian group algebra endomorphism ring Group theory homomorphism ring torsion

Authors and affiliations

  1. 1.Mathematisches InstitutUniversität MünchenMünchenGermany
  2. 2.Department of MathematicsUniversity of HawaiiHonoluluUSA

Bibliographic information

  • Book Title Rings, Modules, and the Total
  • Authors Friedrich Kasch
    Adolf Mader
  • Series Title Frontiers in Mathematics
  • Series Abbreviated Title FM
  • DOI
  • Copyright Information Birkhäuser Basel 2004
  • Publisher Name Birkhäuser Basel
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-7643-7125-8
  • eBook ISBN 978-3-7643-7801-1
  • Series ISSN 1660-8046
  • Series E-ISSN 1660-8054
  • Edition Number 1
  • Number of Pages X, 138
  • Number of Illustrations 3 b/w illustrations, 0 illustrations in colour
  • Topics Algebra
    Associative Rings and Algebras
    Group Theory and Generalizations
  • Buy this book on publisher's site


From the reviews:

“The book is self-contained, well organized and nicely written, making it a very effective introduction to the subject at hand: the total.” (MAA REVIEWS)