Near-rings associated with matched pairs on ring modules

Let G be a module over a ring R, let $\mathcal{C} = \{ {C_i}\}, i \in I$, be a family of submodules of G, and let $\mathcal{H} = \{ {H_i}\}, i \in I$, where ${H_i}$ is a subgroup of $\operatorname{Hom}_R({C_i},G)$ with certain properties. To each such pair $(\mathcal{C},\mathcal{H})$, a near-ring $M(\mathcal{C},\mathcal{H})$ is associated, which is a generalization of the near-ring of homogeneous functions determined by (G, R). The transfer of information from module properties of ${G_R}$ reflected in $(\mathcal{C},\mathcal{H})$ to structural properties of $M(\mathcal{C},\mathcal{H})$ is investigated.