Graph-Induced by Modules via Tensor Product

Abstract

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.