This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R) of a ring R. We establish that vertices of G(R) correspond to modules, with edges defined by the vanishing of their tensor product. Key results include the graph’s connectivity, a diameter of at most 3, and a girth of at most 7 when cycles are present. We show that the set of modules S (R) is empty if and only if R is a field, and that for semisimple rings, the diameter is at most 2. The paper also discusses module isomorphisms over subrings and localization, as well as the inclusion of G(T ) within G(R) for a quotient ring T, highlighting that the reverse inclusion is not guaranteed. Finally, we provide an example illustrating that a non-finitely generated module M does not imply M ⊗M = 0 . These findings deepen our understanding of the interplay among rings, modules, and graphs.
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