Comaximal graph of amalgamated algebras along an ideal
Abstract
Let RR and SS be commutative rings with identity, JJ be an ideal of SS, and let f:R→Sf:R→S be a ring homomorphism. The amalgamation of RR with SS along JJ with respect to ff denoted by R⋈fJR⋈fJ was introduced by D’Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of RR which are transferred to the comaximal graph of R⋈fJR⋈fJ, and also we study some algebraic properties of the ring R⋈fJR⋈fJ by way of graph theory. The comaximal graph of RR, Γ(R)Γ(R), was introduced by Sharma and Bhatwadekar in 1995. The vertices of Γ(R)Γ(R) are all elements of RR and two distinct vertices aa and bb are adjacent if and only if Ra+Rb=RRa+Rb=R. Let Γ2(R)Γ2(R) be the subgraph of Γ(R)Γ(R) generated by non-unit elements, and let J(R)J(R) be the Jacobson radical of RR. It is shown that the diameter of the graph Γ2(R)∖J(R) is equal to the diameter of the graph Γ2(R⋈fJ)∖J(R⋈fJ), and the girth of the graph Γ2(R)∖J(R) is equal to the girth of the graph Γ2(R⋈fJ)∖J(R⋈fJ), provided some special conditions.
Communicated by A. Facchini