some results on a supergraph of the comaximal ideal graph of a commutative ring

The rings considered in this article are commutative with identity whichadmit at least two maximal ideals. we denote the set of all maximal ideals of a ring r by max(r) and we denote the jacobson radical of r by j (r). let r be a ring such that |max(r)| ≥ 2. let i(r) denote the set of all proper ideals of r. in this article, we associate an undirected graph denoted by inc(r) with a subcollection of ideals of r whose vertex set is {i ∈ i(r)|i /⊆ j (r)} and two distinct vertices i₁, i₂ are adjacent in inc(r) if and only if i₁ /⊆ i₂ and i₂ /⊆ i₁ (that is, i₁ and i₂ are not comparable under the inclusion relation). the aim of this article is to investigate the interplay between the graph-theoretic properties of inc(r)