Let \(S=\sum^ n_{i=1} Ra_ i\) be a finite normalizing extension of a ring \(R\) and let \(_ SM\) be a left \(S\)-module. The author proves that the dual Goldie dimension of \(_ RM\) is \(\leq n\) times the dual Goldie dimension of \(_ SM\), provided either \(_ SM\) is Artinian or the group morphism \(M\to a_ i M\) given by \(x\mapsto a_ i x\) is an isomorphism for each \(i\), \(1\leq i\leq n\).