# On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive characteristic

## 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]{approx_equal}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.

- Authors:

- Department of Chemistry, Dnepropetrovsk National University, Dnepropetrovsk (Ukraine)

- Publication Date:

- OSTI Identifier:
- 21267006

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 197; Journal Issue: 9; Other Information: DOI: 10.1070/SM2006v197n09ABEH003803; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; FIELD THEORIES; GROUP THEORY; INTEGRALS; TORSION

### Citation Formats

```
Tushev, A V.
```*On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive characteristic*. United States: N. p., 2006.
Web. doi:10.1070/SM2006V197N09ABEH003803; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Tushev, A V.
```*On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive characteristic*. United States. doi:10.1070/SM2006V197N09ABEH003803; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Tushev, A V. Tue .
"On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive characteristic". United States. doi:10.1070/SM2006V197N09ABEH003803; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21267006,
```

title = {On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive characteristic},

author = {Tushev, A V},

abstractNote = {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]{approx_equal}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.},

doi = {10.1070/SM2006V197N09ABEH003803; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 9,

volume = 197,

place = {United States},

year = {2006},

month = {10}

}