On Divisor Topology of Modules over Domains

TL;DR

This paper introduces a divisor topology on the set of nonzero nongenerators of a module over a domain, extending existing topologies, and explores its properties and applications to various classes of modules.

Contribution

It constructs and analyzes a new divisor topology on modules, extending the divisor topology from domains, and characterizes module classes via this topology.

Findings

D(M) satisfies various separation axioms and countability properties.

D(M) is a Baire space for factorial modules.

Characterization of modules like uniserial, simple, and finitely cogenerated via D(M).

Abstract

Let $M\ $be a module over a domain $R$ and $M^{\#}=\{0\neq m\in M:Rm\neq M\}$ be the set of all nonzero nongenerators of $M.\ $Consider following equivalence relation $\sim$ on $M^{\#}$ as follows: for every $m,n\in M^{\#},\ m\sim n$ if and only if $Rm=Rn.\ $Let $EC(M^{\#})$ be the set of all equivalence classes of $M^{\#}$ with respect to $\sim$. In this paper, we construct a topology on $EC(M^{\#})$ which is called divisor topology of $M\ $and denoted by $D(M).$ Actually, $D(M)$ is extension of the divisor topology $D(R)$ over domains in the sense of Yi\u{g}it and Koc to modules. We investigate separation axioms $T_{i}$ for every $0\leq i\leq5,$ first and second countability, connectivity, compactness, nested property, and Noetherian property on $D(M)$. Also, we characterize some important classes of modules such as uniserial modules, simple modules, vector spaces, and finitely cogenerated modules in terms of $D(M)$. Furthermore, we prove that $D(M)$ is a Baire space for factorial modules. Finally, we introduce and study pseudo simple modules which is a new generalization of simple modules, and use them to determine when $D(M)$ is a discrete space.