Comaximal graph of amalgamated algebras along an ideal

Abstract

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