Comaximal graph of amalgamated algebras along an ideal
Abstract
Let and be commutative rings with identity, be an ideal of , and let be a ring homomorphism. The amalgamation of with along with respect to denoted by was introduced by D’Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of which are transferred to the comaximal graph of , and also we study some algebraic properties of the ring by way of graph theory. The comaximal graph of , , was introduced by Sharma and Bhatwadekar in 1995. The vertices of are all elements of and two distinct vertices and are adjacent if and only if . Let be the subgraph of generated by non-unit elements, and let be the Jacobson radical of . It is shown that the diameter of the graph is equal to the diameter of the graph , and the girth of the graph is equal to the girth of the graph , provided some special conditions.
Communicated by A. Facchini