Regularity and Substructures of Hom

1. Front Matter

   Pages i-xv

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2. Notation and Background

   Pages 1-9

3. Regular Homomorphisms

   Pages 11-40

4. Indecomposable Modules

   Pages 41-48

5. Regularity in Modules

   Pages 49-57

6. Regularity in Hom_R(A, M) as a One-sided Module

   Pages 59-67

7. Relative Regularity: U-Regularity and Semiregularity

   Pages 69-79

8. Reg(A, M) and Other Substructures of Hom

   Pages 81-102

9. Regularity in Homomorphism Groups of Abelian Groups

   Pages 103-129

10. Regularity in Categories

    Pages 131-156

11. Back Matter

    Pages 157-164

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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.

• Abelian group
• algebra
• domain decomposition
• homomorphism
• module category
• regular homomorphism

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)

• Mathematisches Institut, Universität München, München

  Friedrich Kasch

• Department of Mathematics, University of Hawaii, Honolulu, USA

  Adolf Mader

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