Endomorphism Rings of Abelian Groups

1. Front Matter

   Pages i-xii

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2. General Results on Endomorphism Rings

   • Piotr A. Krylov, Alexander V. Mikhalev, Askar A. Tuganbaev

   Pages 1-64

3. Groups as Modules over Their Endomorphism Rings

   • Piotr A. Krylov, Alexander V. Mikhalev, Askar A. Tuganbaev

   Pages 65-98

4. Ring Properties of Endomorphism Rings

   • Piotr A. Krylov, Alexander V. Mikhalev, Askar A. Tuganbaev

   Pages 99-134

5. The Jacobson Radical of the Endomorphism Ring

   • Piotr A. Krylov, Alexander V. Mikhalev, Askar A. Tuganbaev

   Pages 135-180

6. Isomorphism and Realization Theorems

   • Piotr A. Krylov, Alexander V. Mikhalev, Askar A. Tuganbaev

   Pages 181-252

7. Hereditary Endomorphism Rings

   • Piotr A. Krylov, Alexander V. Mikhalev, Askar A. Tuganbaev

   Pages 253-370

8. Fully Transitive Groups

   • Piotr A. Krylov, Alexander V. Mikhalev, Askar A. Tuganbaev

   Pages 371-412

9. Back Matter

   Pages 413-443

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Every Abelian group can be related to an associative ring with an identity element, the ring of all its endomorphisms. Recently the theory of endomor­ phism rings of Abelian groups has become a rapidly developing area of algebra. On the one hand, it can be considered as a part of the theory of Abelian groups; on the other hand, the theory can be considered as a branch of the theory of endomorphism rings of modules and the representation theory of rings. There are several reasons for studying endomorphism rings of Abelian groups: first, it makes it possible to acquire additional information about Abelian groups themselves, to introduce new concepts and methods, and to find new interesting classes of groups; second, it stimulates further develop­ ment of the theory of modules and their endomorphism rings. The theory of endomorphism rings can also be useful for studies of the structure of additive groups of rings, E-modules, and homological properties of Abelian groups. The books of Baer [52] and Kaplansky [245] have played an important role in the early development of the theory of endomorphism rings of Abelian groups and modules. Endomorphism rings of Abelian groups are much stu­ died in monographs of Fuchs [170], [172], and [173]. Endomorphism rings are also studied in the works of Kurosh [287], Arnold [31], and Benabdallah [63].

• Abelian group
• algebra
• endomorphism ring
• torsion

From the reviews:

"In this book, the variety of methods, the beauty of the results, and the appeal of the open problems provide ample justification for the author’s rationale. … book is largely self-contained … . Helpful and challenging sets of exercises appear at the end of each section and open problems are listed at the end of each chapter. This work will be valuable as a text book for a graduate course, a reference on a lively area of mathematics, and a guide to appealing research problems." (Charles Vinsonhaler, Zentralblatt MATH, Vol. 1044 (19), 2004)

• Tomsk State University, Tomsk, Russia

  Piotr A. Krylov

• Moscow State University, Moscow, Russia

  Alexander V. Mikhalev

• Moscow Power Engineering Institute, Technological University, Moscow, Russia

  Askar A. Tuganbaev

Askar Tuganbaev received his Ph.D. at the Moscow State University in 1978 and has been a professor at Moscow Power Engineering Institute (Technological University) since 1978. He is the author of three other monographs on ring theory and has written numerous articles on ring theory.

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