On the Comaximal (Ideal) Graph Associated With Amalgamated Algebra

Abstract

Let f : A → B be a ring homomorphism of the commutative rings A and B with identities. Let J be an ideal of B . The amalgamation of A with B along J with respect to f is a subring of A × B given by A (cid:46)(cid:47) f J := { ( a, f ( a )+ j ) | a ∈ A , j ∈ J } . In this paper, we investigate the comaximal ideal graph and the comaximal graph of the amalgamated algebra A (cid:46)(cid:47) f J . In particular, we determine the Jacobson radical of A (cid:46)(cid:47) f J , characterize the diameter of the comaximal ideal graph of A (cid:46)(cid:47) f J , and investigate the clique number as well as the chromatic number of this graph.