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 of the group algebra of an Abelian torsion-free group of finite rank over a field . The main idea is that the quotient ring by the given ideal can be embedded as an integral domain into some field and the group becomes a subgroup of the multiplicative group of the field . 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 . In turn, the properties of the integral domain depend essentially on the properties of the ideal . 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 of an Abelian torsion-free group of finite rank is obtained in the case where the field has positive characteristic.