How modules and nowhere-zero flows

Let Gamma be a graph, A an abelian group, D a given orientation of Gamma and R a unital subring of the endomorphism ring of A. It is shown that the set of all mappings phi (p from E(Gamma) to A such that (D, phi) is an A-flow forms a left R-module. Let Gamma be a union of two subgraphs Gamma(1) and Gamma(2), and p(n) a prime power. It is proved that Gamma admits a nowhere-zero p(n)-flow if Gamma(1) and Gamma(2) have at most p(n) - 2 common edges and both admit nowhere-zero p(n)-flows. Moreover, it is proved that r admits a nowhere-zero 4-flow if Gamma(1) and Gamma(2) both have nowhere-zero 4-flows and their common edges induce a subgraph of size at most 2 or a connected subgraph of size 3. This result can be seen as a generalization of a theorem of Catlin that a graph admits a nowhere-zero 4-flow if it is a union of a cycle of length at most 4 and a subgraph admitting a nowhere-zero 4-flow.