Endomorphism Rings of Abelian Groups PDF

by P.A. Krylov, Alexander V. Mikhalev, A.A. Tuganbaev

Part of the Algebra and Applications series

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Description

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].