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On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive
characteristic
A. V. Tushev Dnepropetrovsk State University
Abstract:
In the present paper certain methods are developed that enable one
to study the properties of the controller of a prime faithful
ideal
$I$ of the group algebra
$kA$ of an Abelian torsion-free
group
$A$ of finite rank over a field
$k$. The main idea is that
the quotient ring
$kA/I$ by the given ideal
$I$ can be embedded as
an integral domain
$k[A]$ into some field
$F$ and the group
$A$
becomes a subgroup of the multiplicative group of the field
$F$.
This allows one to apply certain results of field theory, such as
Kummer's theory and the properties of the multiplicative groups of
fields, to the study of the integral domain
$k[A]$. In turn, the
properties of the integral domain
$k[A]\cong kA/I$ depend
essentially on the properties of the ideal
$I$. In particular, by
using these methods, an independent proof of the new version of
Brookes's theorem on the controllers of prime ideals of the group
algebra
$kA$ of an Abelian torsion-free group
$A$ of finite rank
is obtained in the case where the field
$k$ has positive
characteristic.
Bibliography: 19 titles.
UDC:
512.544
MSC: Primary
16S34,
20C07; Secondary
11R27,
13B22,
13B30,
13E05,
13F30,
20C15,
20K15 Received: 16.08.2005
DOI:
10.4213/sm1133
Fulltext:
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