Total Graphs of Idealization

Abstract

Let R be a commutative ring with Z(R), its set of zero divisors. The total zero divisor graph of R, denoted Z(Γ(R)) is the undirected (simple) graph with vertices Z(R)=Z(R)-{0}, the set of nonzero zero divisors of R. and for distinct x, y z(R), the vertices x and y are adjacent if and only if x + y z(R). In this paper prove that let R is commutative ring such that Z(R) is not ideal of R then Z((R(+)M)) is connected with diam(Z((R(+)M))) = 2 and the sub graphs Z((R(+)M)) and Reg((R(+)M)) of T((R(+)M)) are not disjoint. And also prove that let R be a commutative ring such that Z(R) is not an ideal of R with Z(R(+)M) = Z(R)(+)M and Reg(R(+)M) = Reg(R)(+)M then Z((R(+)M)) is connected if and only if Z((R) is connected and Reg((R(+)M)) is connected if and only if Reg((R).