Abstract
Let Gamma = (V,E) be a finite undirected graph without loops or multiple edges. A non-empty set of vertices S subset of V is called defensive alliance if every vertex in S have at least one more neighbors inside of S than it has outside of S. A non-empty set of vertices S subset of V is called a dominating set if every vertex not in S is adjacent to at least one member of S. A defensive alliance dominating set is called global. The global defensive alliance number gamma a(Gamma) is defined as the minimum cardinality among all global defensive alliances. In this paper, we initiate the study of the global defensive alliance number of zero-divisor graphs Gamma(R) with R is a finite commutative ring. Hence, we calculate gamma(a)(Gamma(R)) for some usual kind of rings. We finish by a complete characterization of rings with gamma(a)(Gamma(R)) = 1,or 2.