On controllers of prime ideals in group algebras of torsion-free abelian groups of finite rank

Let RA be a group ring of an abelian group A and let I be an ideal of RA . We say that a subgroup B of A controls I if I = (I ∩ RB)RA. The intersection c(I) of all subgroups of A controlling I is said to be the controller of the ideal I ; c(I) is the minimal subgroup of A which controls the ideal I . The ideal I is said to be faithful if I = A ∩ (1 + I) = 1. In theorem 4 we consider some methods for studying of controllers of prime ideals in group algebras of abelian minimax groups. Using these methods, in theorem 6 we obtain an independent proof of a Brookes theorem [1, theorem A] in the case of the field of characteristic zero. In the proofs we use methods which were introduced in [6] and developed in [5, 7]. We will say that a field k is regular if it is countable and the multiplicative group of the field k is a direct product of a torsion group and a free abelian group.