Author(s)

Mohammad Jarrar

Department of Applied Mathematics, Palestine Technical University-Kadoorie, Tulkarem, Palestine.

This paper investigates the connections between ring theory, module theory, and graph theory through the graph $G\left(R\right)$ of a ring R. We establish that vertices of $G\left(R\right)$ 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 $S\left(R\right)$ 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 $G\left(T\right)$ within $G\left(R\right)$ 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 $M\otimes M=0$ . These findings deepen our understanding of the interplay among rings, modules, and graphs.

Graph Theory, Commutative Ring, Tensor Product, Connected, Diameter, Semisimple Ring

Jarrar, M. (2024) Graph-Induced by Modules via Tensor Product. Applied Mathematics, 15, 840-847. doi: 10.4236/am.2024.1512048.