Group Rings and Their Augmentation Ideals
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715 Inder Bir S. Passi
Group Rings and Their Augmentation Ideals
Springer-Verlag Berlin Heidelberg New York 1979
Author Inder Bir S. Passi Department of Mathematics Kurukshetra University Kurukshetra (Haryana) India 132119
AMS Subject Classifications (1970): 16A26, 20C05 ISBN 3-540-09254-4 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09254-4 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 publishel © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
INTRODUCTION The p u r p o s e of these Notes augmentation
is to r e p o r t on some aspects of the
ideal of a group ring.
E v e r y two-sided
ideal in a group ring d e t e r m i n e s a normal
One of the oldest and most c h a l l e n g i n g p r o b l e m s i d e n t i f i c a t i o n of the normal
subgroups,
in group rings
ideal of a group ring.
The first five C h a p t e r s of these N o t e s are devoted that,
to this problem.
subgroups w e r e first studied by M a g n u s
for free groups,
[48] who proved
the integral d i m e n s i o n subgroup series coincides
with the lower central series. More than thirty years demonstrated
is the
called d i m e n s i o n subgroups,
d e t e r m i n e d by the p o w e r s of the a u g m e n t a t i o n Dimension
subgroup.
that Magnus'
later Rips
theorem cannot be g e n e r a l i z e d
[78]
to a r b i t r a r y
groups. Sandling
([79],
[80])
initiated the study of d i m e n s i o n subgroups
over a r b i t r a r y c o e f f i c i e n t rings and of Lie d i m e n s i o n subgroups. a s s o c i a t i v e and the Lie p o w e r s nomial
ideals
[59]. We,
of p o l y n o m i a l the study of
[79] of an a u g m e n t a t i o n
therefore,
begin
ideal are poly-
in Chapter I w i t h the study
ideals. The main result proved (Lie)
Both the
in Chapter II is that for
d i m e n s i o n subgroups over a r b i t r a r y c o e f f i c i e n t rings
it is enough to r e s t r i c t to integral and m o d u l a r
coefficients.
Chapter
III is an e x p o s i t i o n of a p o r t i o n of a fundamental paper of H a r t l e y [27]. Our aim here work [36])
in w h i c h
and Lazard ([35],
([35],
[38] for the study of a u g m e n t a t i o n powers. D i m e n s i o n
fields h a v e been studied b y Bovdi
[36]) , Lazard
calculate these subgroups
[38], Sandling in Chapter
nomial m a p s on groups and c a l c u l a t e
[7], Hall
[26], Jen-
[80] and Z a s s e n h a u s
[99]. We
IV. We study in Chapter V polyintegral d i m e n s i o n subgroups
certain ca
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