Rings, Modules, and the Total

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.

List of contents

Preface.- General Background.- I. Fundamental Notions and Properties.- II. Good Conditions for the Total.- III. The Total of Modules with LE-decompositions.- IV. The Total in Torsion-Free Abelian Groups.- Bibliography.- Index.

Summary

Kasch and Mader developed the theory of the total over the last 10–20 years and put it together for the first time in this book.

Additional text

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)

Report

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)

Product details

• Authors: F. Kasch, Friedric Kasch, Friedrich Kasch, A. Mader, Adolf Mader
• Publisher: Springer, Basel
• Languages: English
• Product format: Paperback / Softback
• Released: 06.07.2004
• EAN: 9783764371258
• ISBN: 978-3-7643-7125-8
• No. of pages: 138
• Weight: 424 g
• Illustrations: X, 138 p. 3 illus.
• Series: Frontiers in Mathematics; Frontiers in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Algebra, B, Gruppen und Gruppentheorie, Group Theory, Mathematics and Statistics, Rings (Algebra), Groups & group theory, Group Theory and Generalizations, Associative rings, Associative Rings and Algebras