Characterizations of derivations on incidence algebras by local actions

Let (X,≤) be a locally finite pre-ordered set and R be a commutative ring with unity. In this paper we apply the theory of zero product determined algebras to show that each linear map on the incidence algebra I(X,R) which is derivable at zero is a generalized derivation and every local derivation on I(X,R) is a derivation.