Abstract
In this paper we first prove that a near-ring admits a derivation if and only if it is zero-symmetric. Also, we prove some commutativity theorems for a non-necessarily 3-prime near-ring R with a suitably-constrained derivation d satisfying the condition that d(a) is not a left zero-divisor in R for some a epsilon R. As consequences, we generalize several commutativity theorems for 3-prime near-rings admitting derivations.