On divisor topology of modules over domains

Tekir, ÜNSAL; Yiğit, Uğur; Buğday, MESUT; Koç, SUAT

doi:10.1080/00927872.2026.2633273

On divisor topology of modules over domains

Tekir Ü., Yiğit U., Buğday M., Koç S.

COMMUNICATIONS IN ALGEBRA, vol.2026, pp.1-15, 2026 (SCI-Expanded, Scopus)

Publication Type: Article / Article
Volume: 2026
Publication Date: 2026
Doi Number: 10.1080/00927872.2026.2633273
Journal Name: COMMUNICATIONS IN ALGEBRA
Journal Indexes: Scopus, Science Citation Index Expanded (SCI-EXPANDED), MathSciNet, zbMATH
Page Numbers: pp.1-15
Marmara University Affiliated: Yes

Abstract

Let M be a module over a domain R and $𝑀 # = {0 \neq 𝑚 \in 𝑀 : 𝑅 𝑚 \neq 𝑀}$ be the set of all nonzero nongenerators of M. Consider the following equivalence relation $\sim$ on $𝑀 #$ given by $𝑚 \sim 𝑛$ if and only if $𝑅 𝑚 = 𝑅 𝑛$ for every $𝑚, 𝑛 \in 𝑀 #$ . Let $𝐸 𝐶 (𝑀 #)$ be the set of all equivalence classes of $𝑀 #$ with respect to $\sim$ . In this paper, we construct a topology on $𝐸 𝐶 (𝑀 #)$ which is called the divisor topology of M and is denoted by $𝐷 (𝑀)$ . Actually, $𝐷 (𝑀)$ is an extension of the divisor topology $𝐷 (𝑅)$ over domains to modules in the sense of Yiğit and Koç. We investigate separation axioms $𝑇 𝑖$ for every $0 \leq 𝑖 \leq 5,$ 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.