Frontiers in Mathematics

Regularity and Substructures of Hom

Authors: Kasch, Friedrich, Mader, Adolf

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  • Readable text with new concepts opening new avenues for research
  • Offers known and numerous new results in a self-contained form
  • Features results that have never before been published in book form
  • Presents an extension of the well-known and important concept of regularity
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About this book

Regular rings were originally introduced by John von Neumann to clarify aspects of operator algebras ([33], [34], [9]). A continuous geometry is an indecomposable, continuous, complemented modular lattice that is not ?nite-dimensional ([8, page 155], [32, page V]). Von Neumann proved ([32, Theorem 14. 1, page 208], [8, page 162]): Every continuous geometry is isomorphic to the lattice of right ideals of some regular ring. The book of K. R. Goodearl ([14]) gives an extensive account of various types of regular rings and there exist several papers studying modules over regular rings ([27], [31], [15]). In abelian group theory the interest lay in determining those groups whose endomorphism rings were regular or had related properties ([11, Section 112], [29], [30], [12], [13], [24]). An interesting feature was introduced by Brown and McCoy ([4]) who showed that every ring contains a unique largest ideal, all of whose elements are regular elements of the ring. In all these studies it was clear that regularity was intimately related to direct sum decompositions. Ware and Zelmanowitz ([35], [37]) de?ned regularity in modules and studied the structure of regular modules. Nicholson ([26]) generalized the notion and theory of regular modules. In this purely algebraic monograph we study a generalization of regularity to the homomorphism group of two modules which was introduced by the ?rst author ([19]). Little background is needed and the text is accessible to students with an exposure to standard modern algebra. In the following, Risaringwith1,and A, M are right unital R-modules.

Reviews

From the reviews:

“This book is dedicated to generalizations of regularity for an Abelian group … . contains an excellent and detailed exposition of results on all types of regularity in Hom with consequences for modules and rings. It is accessible, with all necessary definitions and proofs, contains also a series of instructive examples. … interest both for students and specialists.” (A. I. Kashu, Zentralblatt MATH, Vol. 1169, 2009)

Table of contents (9 chapters)

Table of contents (9 chapters)

Buy this book

eBook $54.99
price for USA in USD
  • ISBN 978-3-7643-9990-0
  • Digitally watermarked, DRM-free
  • Included format:
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $69.95
price for USA in USD
  • ISBN 978-3-7643-9989-4
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Regularity and Substructures of Hom
Authors
Series Title
Frontiers in Mathematics
Copyright
2009
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Basel
eBook ISBN
978-3-7643-9990-0
DOI
10.1007/978-3-7643-9990-0
Softcover ISBN
978-3-7643-9989-4
Series ISSN
1660-8046
Edition Number
1
Number of Pages
XV, 164
Topics