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 (