Comaximal graph of amalgamated algebras along an ideal.

Let R and S be commutative rings with identity, J be an ideal of S , and let f : R → S be a ring homomorphism. The amalgamation of R with S along J with respect to f denoted by R ⋈ f J was introduced by D'Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of R which are transferred to the comaximal graph of R ⋈ f J , and also we study some algebraic properties of the ring R ⋈ f J by way of graph theory. The comaximal graph of R , Γ (R) , was introduced by Sharma and Bhatwadekar in 1995. The vertices of Γ (R) are all elements of R and two distinct vertices a and b are adjacent if and only if R a + R b = R. Let Γ 2 (R) be the subgraph of Γ (R) generated by non-unit elements, and let J (R) be the Jacobson radical of R. It is shown that the diameter of the graph Γ 2 (R) ∖ J (R) is equal to the diameter of the graph Γ 2 (R ⋈ f J) ∖ J (R ⋈ f J) , and the girth of the graph Γ 2 (R) ∖ J (R) is equal to the girth of the graph Γ 2 (R ⋈ f J) ∖ J (R ⋈ f J) , provided some special conditions. [ABSTRACT FROM AUTHOR]