S-Zariski topology on S-spectrum of modules

Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, first we give some relations between S-prime and S-maximal submodules that are generalizations of prime and maximal submodules, respectively. Then we construct a topology on the set of all S-prime submodules of M , which is generalization of prime spectrum of M. We investigate when SpecS(M) is T0 and T1-space. We also study on some continuous maps and irreducibility on SpecS(M). Moreover, we introduce the notion of S-radical of a submodule N of M and use it to show the irreducibility of S-variety VS(N).