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Mat. Sb., 2006, Volume 197, Number 9, Pages 115–160 (Mi msb1133)  

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: PDF file (721 kB)
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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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\paper On~the controllers of prime ideals of group algebras of~Abelian torsion-free groups of~finite rank over a~field of positive
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\vol 197
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\pages 115--160
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\transl
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\yr 2006
\vol 197
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\pages 1365--1404
\crossref{https://doi.org/10.1070/SM2006v197n09ABEH003803}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Tushev, “On certain methods of studying ideals in group rings of abelian groups of finite rank”, Asian-European J. Math, 2014, 1450065  crossref  mathscinet  zmath
  • Математический сборник Sbornik: Mathematics (from 1967)
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