Quasi Zariski-like topology on quasi-primary spectrum of lattice modules

Abstract

Let L denote a multiplicative lattice and M be a lattice module over L. A proper element K of a lattice module M is called quasi-primary, whenever the relation aA ≤ K implies either a ≤ √(K : 1_M) or A ≤ √K. In this work, we define the set Spec^qp(M) as the family of all quasi-primary elements K of M satisfying the condition (√K : 1_M) = √(K : 1_M) and this collection is referred to as the quasi-primary spectrum. Furthermore, we establish Zariski-type topology on quasi-primary spectrum. We then examine several topological characteristics of this space, with particular emphasis on compactness and irreducibility. In addition, we prove that the space is T₀ and spectral whenever the set Spec^qp_p(M) = {K ∈ Spec^qp(M) | √(K : 1_M) = p, p ∈ Spec(L)} consists of a single element.

1. Introduction

The study of Zariski-type topologies has been extended beyond the prime spectrum of commutative rings to the prime submodules of modules, leading to several important generalizations in recent years [6, 17]. One of the contributions in this direction was made by Behboodi and Noori [7], who introduced a Zariski-type topology on the classical prime spectrum of modules. Subsequently, Girase et al. [13] adapted this framework to the setting of lattice modules by developing a corresponding topology on their classical prime spectrum. Likewise, Borkar et al. [8] established a Zariski-type topological structure for the prime spectrum associated with lattice modules. Furthermore, M. Samiei et al. [21] introduced the notion of quasi-primary submodules and investigated the topological properties of their spectrum. Inspired by these developments, the present work focuses on the spectrum of quasi-primary elements in a lattice module M over a C-lattice L. We introduce a Zariski-like topology on this spectrum and examine its fundamental properties. In addition, several known results concerning prime and primary spectra are extended to the quasi-primary setting.

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Girase, P. & Phadatare, N. (2026). Quasi Zariski-like topology on quasi-primary spectrum of lattice modules. Asia Mathematika, 10(2), 124–135. https://doi.org/10.5281/zenodo.23123720

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