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This article is cited in 1 scientific paper (total in 1 paper)
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.
DOI:
https://doi.org/10.4213/sm1133
Full text:
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English version:
Sbornik: Mathematics, 2006, 197:9, 1365–1404
Bibliographic databases:
UDC:
512.544
MSC: Primary 16S34, 20C07; Secondary 11R27, 13B22, 13B30, 13E05, 13F30, 20C15, 20K15 Received: 16.08.2005
Citation:
A. V. Tushev, “On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive
characteristic”, Mat. Sb., 197:9 (2006), 115–160; Sb. Math., 197:9 (2006), 1365–1404
Citation in format AMSBIB
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Linking options:
http://mi.mathnet.ru/eng/msb1133https://doi.org/10.4213/sm1133 http://mi.mathnet.ru/eng/msb/v197/i9/p115
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This publication is cited in the following articles:
-
A. V. Tushev, “On certain methods of studying ideals in group rings of abelian groups of finite rank”, Asian-European J. Math, 2014, 1450065
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