Projective modules over group rings

Abstract

I,et R be a Dedekind ring of characteristic zero and L7 a finite group such that no rational prime dividing the order of rf is invertible in K. Let T be a free Abelian group or monoid of rank m. We prove that for any projective R[T][IT]-module P of rank 2- vz $- 2, 1’ -p F ‘3 K c, 91, where F is a free module, i? a module of rank 1 and ?I is an ideal of R[T][fl] such that ‘?I ~- ‘?I, Qqn, R[T][IT],where 91, is a projective ideal of R[ITj (Theorem 2.2). LVe also prove that for any free Abelian monoid 7’ and a projective R[T][II]- module P, L 821 P is L[T][T7] ~stablv free where L is the quotient field of R(‘l’heorem 2.1) and show that if rank of 7’, i.e., rank 7’ z-m I, then L c P is L[T][n]-free (Corollaq J 2.2); moreover, in this case any projective module P of rank ;.’ 3 is of the form F @, 91, whrrc F is a free module and VI is an ideal of R[T][Uj, this generalizes a similar result in Rcf [5], proved for R Z with X7 a commutative group, except for rank P -~: 2. In the Section 3, we generalize some results about ;V”&(Zl11]) and Li<,,(Z(rf]) of Fief. 131 (see Ref. [I, Chap 121) and deduce that for any projective module P over