Properties of endomorphism rings of Abelian groups. II.
type: Item
instance of: scholarly article
title: Properties of endomorphism rings of Abelian groups. II.
zbMATH Open document ID: 1054.20038
DOI: 10.1023/A:1020880708517
published in: Journal of Mathematical Sciences (New York)
publication date: 2003-05-29T00:00:00Z
review text: This paper contains the final three chapters of what is now a book: Endomorphism Rings of Abelian Groups [Algebras and Applications 2. Boston MA: Kluwer Academic Publishers (2003; Zbl 1044.20037)]. The first four chapters were published as a separate paper [part I, J. Math. Sci., New York 112, No. 6, 4598-4735 (2002; Zbl 1041.20039)]. Given an Abelian group \(A\), we can form the ring of endomorphisms \(\text{End}_\mathbb{Z}(A)\) and then view the group \(A\) as an \(\text{End}_\mathbb{Z}(A)\)-module in a natural way. A wide variety of tools are available for studying such objects, and the reader of ``Endomorphism Rings'' is assumed to have some familiarity with the theories of Abelian groups, rings, and modules. Anyone with the prerequisites will enjoy the variety of methods, the beauty of the results, and the appeal of the open problems compiled by the authors. As background, I'll list the first four chapters of the book: Chapter I (General Results on Endomorphism Rings). Chapter II (Groups as Modules over Their Endomorphism Rings). Chapter III (Ring Properties of Endomorphism Rings). Chapter IV (Jacobson Radical). Chapter V (Isomorphism or Realization Theorems for Endomorphism Rings) is a discussion of the relationship between two groups whose endomorphism rings are isomorphic, and of which rings can be ``realized'' as endomorphism rings of various kinds of groups. The Baer-Kaplansky Theorem for torsion Abelian groups and Corner's Theorem on countable, reduced torsion-free rings are perhaps the most well-known results in these areas. Due to the limitations of space, the chapter omits the extensive literature on realizing rings as endomorphism rings of ``large'' Abelian groups using the Black Box of Shelah and related combinatorial arguments. Chapter VI (Hereditary Endomorphism Rings and Related Topics). A ring \(R\) is called right hereditary if each right ideal is projective as a right \(R\)-module. The importance of this property for endomorphism rings of Abeli…
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1916225
zbMATH Keywords: Abelian groups
zbMATH Keywords: endomorphism rings
zbMATH Keywords: additive groups of rings
zbMATH Keywords: quasi-endomorphisms
zbMATH Keywords: Jacobson radical
author: P. A. Krylov
author: A. V. Mikhalev
author: A. A. Tuganbaev
reviewed by: Charles Vinsonhaler
MaRDI profile type: Publication
full work available at URL: a:1020880708517
OpenAlex ID: W4245996968
description: scientific article; zbMATH DE number 1916225