Localization and Topological Aspects of Quasi-Prime Submodules

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] a multiplicative subset of [Formula: see text]. A submodule [Formula: see text] of an [Formula: see text]-module [Formula: see text] with [Formula: see text] is called a quasi-[Formula: see text]-prime submodule if there exists a fixed [Formula: see text], and whenever [Formula: see text] for some [Formula: see text], then [Formula: see text] or [Formula: see text]. The set of all quasi-[Formula: see text]-prime submodules of [Formula: see text] is called the localization of Spectrum of [Formula: see text] by [Formula: see text] and denoted by [Formula: see text]. In this paper, we construct and study a topology on [Formula: see text]. We investigate some connections between algebraic properties of [Formula: see text] and topological properties of [Formula: see text] such as separation axioms, compactness, connectedness and irreducibility.