S-Second classical S-Zariski topology of a module

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a multiplicatively closed subset of [Formula: see text]. A submodule [Formula: see text] of an [Formula: see text]-module [Formula: see text] with ann[Formula: see text] is said to be an [Formula: see text]-second submodule if there exists a fixed [Formula: see text], and whenever [Formula: see text], where [Formula: see text] and [Formula: see text] is a submodule of [Formula: see text], we have either [Formula: see text] or [Formula: see text]. The set of all [Formula: see text]-second submodules of an [Formula: see text]-module [Formula: see text] will be called [Formula: see text]-second spectrum of [Formula: see text] and denoted by [Formula: see text]-[Formula: see text]. In this paper, we construct and investigate a topology on [Formula: see text]-[Formula: see text] which we will call [Formula: see text]-second classical [Formula: see text]-Zariski topology. We examine separation axioms, compactness, connectedness and irreducibility of this topological space. We also investigate [Formula: see text]-second classical [Formula: see text]-Zariski topology from the point of view spectral spaces by using Hochster’s characterization.