Rings and Semigroups
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380 Mario Petrich Pennsylvania State University, University Park, PA/USA
With an Appendix by Richard Wiegandt
Rings and Semigroups
Springer-Verlag Berlin· Heidelberg· New York 1974
AMS Subject Classifications (1970): 20·02, 20-M-20, 20-M-25, 20-M-30, 22-A-30, 16·02, 16A12, 16A20, 16A42, 16A56, 16A64, 16A80
ISBN 3·540·06730·2 Springer-Verlag Berlin· Heidelberg· New York ISBN 0-387·06730·2 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 photo· copying 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 the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin · Heidelberg 1974. Library of Congress Catalog Card Number 74·2861. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
INTRODUCTION
Semigroup theory can be considered as one of the more ring theory.
successf~l
offsprings of
The relationship of these two theories has been a subject of particu-
lar attention only within the last two decades and has generally taken the form of an investigation of the multiplicative semigroups of rings.
The first and still
the most fundamental work in this direction is due to L.M. Gluskin who studied certain dense rings of linear transformations from the multiplicative point of view. These Lectures represent an attempt to put selected topics concerning both rings of linear transformations and abstract rings, as well as their multiplicative semigroups, into a form suitable for presentation to students interested in algebra. The Lectures are divided into three parts according to the clusters of covered topics. Part I consists of a study of certain semigroups and rings of linear transformations on an arbitrary vector space over a division ring.
For dense rings of
linear transformations containing a nonzero linear transformation of finite rank, two phenomena, from the present point of view, are of decisive importance:
(a) its
multiplicative semigroup is a dense extension of a completely 0-simple semigroup, and
(b) it has unique addition.
Because of (b), most information about these rings
can be obtained by considering their multiplicative semigroups alone.
This leads
naturally to a st•1dy of semigroups of linear transformations and in particular of those satisfying (a) above.
Hence in many instances we first establish the desired
result for semigroups and then specialize it to rings of linear transformations.
[4),
Even though the guiding idea adopted here is that first expounded by Gluskin the principal references are the books of Baer [l) and Jacobson
[7).
Part II contains an investigation of various abstract rings, izations and representations.
their character-
The classes of rings under study here are semiprime
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