Domination Number of Nagata Extension Ring

Let R is a commutative ring whit Z(R) the set of zero divisors. The total graph of R, denoted by T(Γ(R)) is the (undirected) graph with all elements of R as vertices, and two distinct vertices are adjacent if their sum is a zero divisor. For a graph G = (V, E), a set S is a dominating set if every vertex in V (back slash) S is adjacent to a vertex in S. The domination number is equal |S| where |S| is minimum. For R-module M, an Nagata extension (idealization), denoted by R(+)M is a ring with identity and for two elements (r, m),(s, n) of R(+)M we have (r, m)+ (s, n) = (r+s, m+n) and (r, m)(s, n) = (rs, rn+sm). In this paper, we seek to determine the bound for the domination number of total graph T(Γ(R(+)M)).