A New Extension of Comaximal Graph of a Commutative Ring

A New Extension of Comaximal Graph of a Commutative Ring

Let R be a commutative ring without identity. We define the graph G(R) with vertex set V (G(R)) and edge set E(G(R)) as follows: V (G(R)) ={|II ̸= {0}, I ◁ R} E(G(R)) = {IJ|I+J + J = R} The set △(R) consists of all ideals I of R such that I is not contained in J(R), where J(R) denotes the Jacobson radical of R. Throughout this paper we consider only commutative ring not necessary unital. In this paper we study about this graph. We show that under some conditions on the G(R), the ring R is Noetherian or Artinian.