Rings and Categories of Modules

1. Front Matter

   Pages i-ix

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2. Preliminaries

   • Frank W. Anderson, Kent R. Fuller

   Pages 1-9

3. Rings, Modules and Homomorphisms

   • Frank W. Anderson, Kent R. Fuller

   Pages 10-64

4. Direct Sums and Products

   • Frank W. Anderson, Kent R. Fuller

   Pages 65-114

5. Finiteness Conditions for Modules

   • Frank W. Anderson, Kent R. Fuller

   Pages 115-149

6. Classical Ring-Structure Theorems

   • Frank W. Anderson, Kent R. Fuller

   Pages 150-176

7. Functors Between Module Categories

   • Frank W. Anderson, Kent R. Fuller

   Pages 177-249

8. Equivalence and Duality for Module Categories

   • Frank W. Anderson, Kent R. Fuller

   Pages 250-287

9. Injective Modules, Projective Modules, and Their Decompositions

   • Frank W. Anderson, Kent R. Fuller

   Pages 288-326

10. Back Matter

    Pages 327-339

    PDF

This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. We assume the famil­ iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuing theme of the text is the study of the relationship between the one-sided ideal structure that a ring may possess and the behavior of its categories of modules. Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Art in Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, de­ composition theory of injective and projective modules, and semiperfect and perfect rings. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty. There are, of course, many important areas of ring and module theory that the text does not touch upon. For example, we have made no attempt to cover such subjects as homology, rings of quotients, or commutative ring theory.

• Area
• Categories
• Division
• Finite
• Modules
• Rings
• algebra
• behavior
• density
• duality
• functions
• homomorphism
• ring
• theorem
• transformation

• Department of Mathematics, University of Oregon, Eugene, USA

  Frank W. Anderson

• Division of Mathematical Sciences, The University of Iowa, Iowa City, USA

  Kent R. Fuller

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