Fixed Rings of Finite Automorphism Groups of Associative Rings
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818 Susan Montgomery
Fixed Rings of Finite Automorphism Groups of Associative Rings
Springer-Verlag Berlin Heidelberg New York 1980
Author Susan Montgomery Department of Mathematics, University of Southern California Los Angeles, CA 90007/USA
AMS Subject Classifications (1980): 16-02, 16A08, 16A33, 16A34, 16A38, 16A72 ISBN 3-540-10232-9 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10232-9 Springer-Verlag NewYork 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
These notes
are d e d i c a t e d
Ora Beck German
to
PREFACE The first six chapters of these lecture notes are an expanded version of a series of seminar lectures, Southern California
begun at the University of
in the fall semester,
1976, continued at the
University of Chicago in the spring of 1977, the spring of 1978.
and completed at USC in
Chapter 7 includes some more recent results on
skew group rings and modules,
and provides
some alternate proofs of
material presented earlier. The intent of the lectures was to describe some of the many new results on fixed rings of automorphism groups which had been obtained since 1970; a g r e a t deal of progress had been made concerning the relationship of the structure of a ring R to the structure of a fixed subring R G with respect to a finite automorphism group G. to study this relationship, what circumstances teed.
In order
it was first necessary to know under
the existence of fixed elements
could be guaran-
We mention two papers which were fundamental to this problem:
the 1973 paper of G° Bergman and I.M. Isaacs, that if R is semiprime with no
IGI- torsion
order of G), then R G is non-trivial; V.K. Kharchenko,
in which they establish
(where
IGldenotes the
and the first 1975 paper of
in which he shows that if R has no nilpotent
elements and G is any finite group, is in this paper that Kharchenko
then R G is non-trivialo
Also,
it
introduces his notion of generalized
inner automorphisms. Once these results on the existence of R G were known, on chain conditions, became tractable,
polynomial
identities,
and R and R G - modules
in the situation when R had no
had no nilpotent elements, inner automorph-isms.
questions
IGI- torsion,
or R
or the group contained no "generalized"
These questions are the ones discussed here,
Vl along with the existence of trace functions
from R to RG and the rela-
tionship of R G to the skew group ring R*G.
These results occur in
work of M. Cohen, D.
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