Let R be a commutative ring with identity and let S be a multiplicatively closed subset of R. A submodule P of an R-module M with \((P:_{R}M)\cap S=\emptyset \) is said to be an S-prime submodule of M if there exists a fixed \(s\in S\) and whenever \(am\in P\) , then \(sa\in (P:_{R}M)\) or \(sm\in P\) for each \(a\in R\) , \(m\in M\) . The set of all S-prime submodules of M is denoted by \(Spec_{S}(M)\) . In this paper, we construct and investigate a topology on \(Spec_{S}(M)\) which we will call classical S-Zariski topology for an R-module M. We use specific algebraic properties of M to obtain some topological properties such as separation axioms, compactness, connectedness, and irreducibility. We also investigate classical S-Zariski topology from the point of view spectral spaces by using Hochster’s characterization.
