TITLE:
Graph-Induced by Modules via Tensor Product
AUTHORS:
Mohammad Jarrar
KEYWORDS:
Graph Theory, Commutative Ring, Tensor Product, Connected, Diameter, Semisimple Ring
JOURNAL NAME:
Applied Mathematics,
Vol.15 No.12,
December
9,
2024
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.