Associated Prime Ideals of the Amalgamated Algebra

Let R and S be commutative rings with identity, J an ideal of S, and f : R → S a ring homomorphism. In this paper, we determine the set of associated prime ideals of amalgamated algebra R f J , when f is surjective. As an application, we characterize when the zero-divisor graph of amalgamations is a (complete) bipartite graph.