Planar Torsion Graph of Modules

Abstract

Let $R$ be a commutative ring with identity, $M$ be an $R$-module and $T(M)^*$ be the set of
nonzero torsion elements. The set $T(M)^*$ makes up the vertices of the corresponding torsion graph,
$\Gamma_{R}(M)$, with two distinct vertices $x, y\in T(M)^*$ forming an edge if $Ann(x)\cap Ann(y)\neq 0$.
In this paper we study the case where the torsion graph $\Gamma_{R}(M)$ is planar. We show that if the torsion graph of a
multiplication $R$-module $M$ is planar, then $M$ is both Noetherian and Artinian. We also study the relation-
ship between the planar torsion graph and the maximal submodule of a module.