ZERO-DIVISOR GRAPHS OF AMALGAMATIONS

Let f : A → B be a homomorphism of commutative rings and let J be an ideal of B. The amalgamation of A with B along J with respect to f is the subring of A × B given by A ⋈f J := {(a, f(a) + j) | a ∊ A, j ∊ J}. This paper investigates the zero-divisor graph of amalgamations. Our aim is to characterize when the graph is complete and compute its diameter and girth for various contexts of amalgamations. The new results recover well-known results on duplications, and yield new and original examples issued from amalgamations.