In [4], Macaulay proved that μ(mI) ≥Q(μ(I)) ≥ μ(I) for any homogeneous idealIink[X1, …,Xn] with minimal generators of a single degree wheremis the homogeneous maximal ideal (X1, …,Xn) andQis the function called the Binomial Expansion basen- 1. The inequelity can be generalised to the case of torsion-fre modules with generators of a single degree. The proof follows the lines of Robbiano [5]. A family of counterexamples to μ(mI) ≥ μ(I) in whichIhas minimal generators of just two different degrees is given.