Quasi-Zariski Topology on the Quasi-Primary Spectrum of A Module

Abstract

Let R be a commutative ring with a nonzero identity and M be a unitary R-module. A submodule Q of M is called quasi-primary if Q ≠ M and, whenever r ∈ R, x ∈ M, and rx ∈ Q, we have r ∈ √(Q : M) or x ∈ radQ. A submodule N of M satisfies the primeful property if and only if M/N is a primeful R-module. We let q.Spec(M) denote the set of all quasi-primary submodules of M satisfying the primeful property. The aim of this paper is to introduce and study a topology on q.Spec(M) which is called quasi-Zariski topology of M. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration. Modules whose quasi-Zariski topology is, respectively T₀, T₁ or irreducible, are studied, and several characterizations of such modules are given. Finally, we obtain conditions under which q.Spec(M) is a spectral space.

Key words and phrases. Quasi-primary submodule, quasi-primaryful module, quasi-Zariski topology, quasi-primary spectrum.

2000 Mathematics Subject Classification. 13C13, 13C99, 54B99