On Golomb Topology of Modules over Commutative Rings

TL;DR

This paper introduces a new Golomb topology on modules over commutative rings, linking topological properties with algebraic structures and characterizing key module classes.

Contribution

It defines the Golomb topology for certain modules and explores its properties, providing characterizations of simple and Jacobson semisimple modules.

Findings

Golomb topology basis is formed by coprime cosets under specific conditions.

Topological properties of G(M) relate to algebraic properties of modules.

Characterizations of simple and Jacobson semisimple modules via Golomb topology.

Abstract

In this paper, we associate a new topology to a nonzero unital module $M$ over a commutative $R$, which is called Golomb topology of the $R$-module $M$. Let $M\ $be an\ $R$-module and $B_{M}$ be the family of coprime cosets $\{m+N\}$ where $m\in M$ and $N\ $is a nonzero submodule of $M\ $such that $N+Rm=M$. We prove that if $M\ $is a meet irreducible multiplication module or $M\ $is a meet irreducible finitely generated module in which every maximal submodule is strongly irreducible, then $B_{M}\ $is the basis for a topology on $M\ $which is denoted by $\widetilde{G(M)}.$ In particular, the subspace topology on $M-\{0\}$ is called the Golomb topology of the $R$-module $M\ $and denoted by $G(M)$. We investigate the relations between topological properties of $G(M)\ $and algebraic properties of $M.\ $In particular, we characterize some important classes of modules such as simple modules, Jacobson semisimple modules in terms of Golomb topology.