Isomorphisms between Jacobson graphs

Abstract

Abstract. Let R be a commutative ring with a non-zero identity and J R be its Jacobson graph. We show that if R and R ′ are finite commutative rings, then J R ∼ = J R ′ if and only if |J(R)| = |J(R ′ )| and R/J(R) ∼ = R ′ /J(R ′ ). Also, for a Jacobson graph J R , we obtain the structure of group Aut(J R ) of all automorphisms of J R and prove that under some conditions two semi-simple rings R and R ′ are isomorphic if and only if Aut(J R ) ∼ = Aut(J R ′ ).