Ebook: Endomorphism Rings of Abelian Groups

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].

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This book is the first monograph on the theory of endomorphism rings of Abelian groups. The theory is a rapidly developing area of algebra and has its origin in the theory of operators of vector spaves. The text contains additional information on groups themselves, introducing new concepts, methods, and classes of groups. All the main fields of the theory of endomorphism rings of Abelian groups from early results to the most recent are covered. Neighbouring results on endomorphism rings of modules are also mentioned.

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This book is the first monograph on the theory of endomorphism rings of Abelian groups. The theory is a rapidly developing area of algebra and has its origin in the theory of operators of vector spaves. The text contains additional information on groups themselves, introducing new concepts, methods, and classes of groups. All the main fields of the theory of endomorphism rings of Abelian groups from early results to the most recent are covered. Neighbouring results on endomorphism rings of modules are also mentioned.
Content:
Front Matter....Pages i-xii
General Results on Endomorphism Rings....Pages 1-64
Groups as Modules over Their Endomorphism Rings....Pages 65-98
Ring Properties of Endomorphism Rings....Pages 99-134
The Jacobson Radical of the Endomorphism Ring....Pages 135-180
Isomorphism and Realization Theorems....Pages 181-252
Hereditary Endomorphism Rings....Pages 253-370
Fully Transitive Groups....Pages 371-412
Back Matter....Pages 413-443

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This book is the first monograph on the theory of endomorphism rings of Abelian groups. The theory is a rapidly developing area of algebra and has its origin in the theory of operators of vector spaves. The text contains additional information on groups themselves, introducing new concepts, methods, and classes of groups. All the main fields of the theory of endomorphism rings of Abelian groups from early results to the most recent are covered. Neighbouring results on endomorphism rings of modules are also mentioned.
Content:
Front Matter....Pages i-xii
General Results on Endomorphism Rings....Pages 1-64
Groups as Modules over Their Endomorphism Rings....Pages 65-98
Ring Properties of Endomorphism Rings....Pages 99-134
The Jacobson Radical of the Endomorphism Ring....Pages 135-180
Isomorphism and Realization Theorems....Pages 181-252
Hereditary Endomorphism Rings....Pages 253-370
Fully Transitive Groups....Pages 371-412
Back Matter....Pages 413-443
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