Let R be a ring, let M be a left R-module, and let τ : M → M and δ : R → R be additive maps. We say that τ is a generalized derivation relative to δ if τ(am) = aτ(m) + δ(a)m for all a ∈ R and m ∈ M. In this paper, we provide a generalization of Posner’s first theorem to generalized derivations on 2-torsion free prime modules. We also obtain a result of this generalization in connection with derivations acting on left ideals of prime rings. Moreover, we extend some previous results related to Posner’s first theorem. Furthermore, as an application of our main result, we examine Posner’s first theorem for a certain
