This paper is concerned with investigating the idea of the finite Goldie dimension (FGD) of a near-ring module. All near-rings \(N\) considered a right near-rings and distributively generated with no independent family of left \(N\) subgroups and with the ascending chain condition on annihilators. In this context many of the concepts used in ring theory connected with FGD are carried over. This enables several results on the rank of \(N\) groups to be obtained, though strong extra hypotheses are needed, in particular that the near-ring is strongly semi-prime (no non-zero nilpotent invariant subset) and left annihilators are distributively generated.