The literatrue on near-rings contains a number of theorems asserting that certain conditions implying commutativity in rings imply multiplicative or additive commutativity in special classes of near-rings. H. E. Bell and G. Mason in [2] added to this body of results several commutativity theorems for near-rings admitting suitably-constrained derivations. In this paper we generalize some of their results to a subclass of prime near-rings admitting suitably-constrained $ \sigma $-derivations, where $ \sigma $ is an automorphibm of the prime near-ring.