THE TOTAL GRAPH OF A COMMUTATIVE RING WITH RESPECT TO PROPER IDEALS

Let R be a commutative ring and 1 its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T(Gamma(I)(R)). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y is an element of R., the vertices x and y are adjacent if and only if x + y is an element of S(I). The total graph of a commutative ring, that denoted by T(Gamma(R)), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x y is an element of Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, T(Gamma(I)(R)) = T(Gamma(R)); this is an important result on the definition.