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

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.
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