Some results on the co-intersection graph of submodules of a module

No abstract

No abstract

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be a unitary [Formula: see text]-module. We associate an undirected simple graph to an [Formula: see text]-module [Formula: see text] denoted by [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists the nonzero element [Formula: see text] in [Formula: see text] such that [Formula: see text] or [Formula: see text]. In this paper, we investigate the interplay between the graph-theoretic properties of [Formula: see text] and algebraic properties of the module [Formula: see text]. We study the connectivity and completeness of [Formula: see text]. Also, the diameter and the girth of this graph are determined. We prove that if [Formula: see text] is an [Formula: see text]-module with [Formula: see text], then [Formula: see text] has a cycle or [Formula: see text], for some [Formula: see text]. Also, we show that if [Formula: see text] is a multiplication [Formula: see text]-module and [Formula: see text] is a planar graph, then every prime submodule of [Formula: see text] is non-faithful.