Additive Derivations of Incidence Modules

Let R be an associative ring and M a left R-module. This paper examines the structural properties of the incidence module $I(\mathcal{P},M)$, associated with a module M over a ring R and a locally finite poset $\mathcal{P}$. We provide a complete characterization of when an additive derivation on $I(\mathcal{P},M)$ is inner, for the case where $\mathcal{P}$ is a finite and connected poset. These criteria are then generalized to arbitrary posets, revealing a profound connection between the algebraic properties of the module and the graph-theoretic structure of $\mathcal{P}$ as a directed graph.