On Graphs Related to Comaximal Ideals of a Commutative Ring

Abstract

We study the co maximal graph Ω ( R ) , the induced subgraph Γ ( R ) of Ω ( R ) whose vertex set is R ∖ ( U ( R ) ∪ J ( R ) ) , and a retract Γ r ( R ) of Γ ( R ) , where R is a commutative ring. For a graph Γ ( R ) which contains a cycle, we show that the core of Γ ( R ) is a union of triangles and rectangles, while a vertex in Γ ( R ) is either an end vertex or a vertex in the core. For a nonlocal ring R , we prove that both the chromatic number and clique number of Γ ( R ) are identical with the number of maximal ideals of R . A graph Γ r ( R ) is also introduced on the vertex set { R x ∣ x ∈ R ∖ ( U ( R ) ∪ J ( R ) ) } , and graph properties of Γ r ( R ) are studied.