Domination Number of Nagata Extension Ring

Abstract

Aََََbstract:Let R is a commutative ring whit Z(R) as the set of zero divisors. The total graph of R, denoted by T ((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 n 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 ((R(+)M)).