Fitting ideals and the Gorenstein property

Abstract

Let p be a prime number and G be a finite commutative group such that p 2 does not divide the order of G. In this note we prove that for every finite module M over the group ring Z p [G], the inequality \({\#M\,\leq\,\#{\bf Z}_{p}[G]/{{\rm Fit}}_{{\bf Z}_{p}[G]}(M)}\) holds. Here, \({\rm Fit}_{{\bf Z}_{p}[G]}(M)\) is the Z p [G]-Fitting ideal of M.