Finite Rank Torsion Free Abelian Groups and Rings
- PDF / 9,518,238 Bytes
- 199 Pages / 461 x 684 pts Page_size
- 99 Downloads / 239 Views
931
David M. Arnold
Finite Rank Torsion Free Abelian Groups and Rings
Springer-Verlag Berlin Heidelberg New York 1982
Author David M. Arnold Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003, USA
AMS Subject Classifications (1980): 20 K 15, 20 K 30, 20 K 40 ISBN 3-540-11557-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11557-9 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
INTRODUCTION
These notes contain a largely expository introduction to the theory of finite rank torsion free abelian groups developed since the publication of "Infinite Abelian Groups," Vol. II, L. Fuchs, in 1973.
As reflected in
Chapter XIII of that text, the subject consists of a satisfactory theory for direct sums of rank 1 groups due to R. Baer in 1937; a uniqueness of quasidirect sum decompositions up to quasiisomorphism due to B. Jonsson in 1959; a realization of subrings of finite dimensional Qalgebras as endomorphism rings due to A.L.S. Corner in 1963; a variety of pathological direct sum decompositions; and some apparently miscellaneous results largely relegated to the exercises. Substantial progress has been made in the subject since 1973.
Most
notable are the stable range conditions proved by R.B. Warfield, near isomorphism as introduced by E.L. Lady, and the application of properties of subrings of finite dimensional Qalgebras to finite rank torsion free abelian groups via a Moritalike duality developed by E.L. Lady and the author.
Consequently,
some older results of R. Beaumont, R. Pierce, and J. Reid (c. 1960) involving subrings of finite dimensional Qalgebras gain new importance.
Thus a sys-
tematic introduction to the theory of finite rank torsion free abelian groups and subrings of finite dimensional Qalgebras seems timely. The theory of direct sums of rankl torsion free abelian groups has been combined with the theory of totally projective groups to characterize a class of mixed abelian groups (Warfield [7] and HunterRichman [1]).
The category Walk, as
discussed in Warfield [ 7 ], has been used to investigate mixed abelian groups. A secondary goal of these notes is to survey the known results for finite rank torsion free abelian groups with an eye towards eventual application to mixed groups of finite torsion free rank via the category Walk.
Some progress
IV
along these lines is reported by Warfield [7
J,
Other potential applications
include the study of mixed abelian groups of finite torsion
Data Loading...
