S-Zariski Topology on S-spectrum of Modules

Abstract

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 $Spec_S(M)$ is $T_0$ and $T_1$-space. We also study on some continuous maps and irreducibility on $Spec_S(M)$. Moreover, we introduce the notion of $S$-radical of a submodule $N$ of $M$ and use it to show the irreducibility of $S$-variety $V_S(N)$.