# Isolated Submodules and Skew Fields

## Document Type

Article

## Publication Date

5-1-2000

## Department

Mathematics

## School

Mathematics and Natural Sciences

## Abstract

We study the generally distinct concepts of isolated submodule, honest submodule, and relatively divisible submodule for unital right *R*-modules, where *R* is an associative ring with identity. This is accomplished by studying a certain subset called the *Q*-torsion subset relative to a subset *Q* (sometimes a right ideal but not always) of *R*. The *Q*-isolator turns out to always to be a categorical closure operator and the notion of *Q*-honest is an `operator" but need not be a closure operator. It is shown that the notions of *Q*-isolated and *Q*-honest coincide precisely when the *Q*-honest operator is a closure operator and this happens precisely when all submodules are *Q*-honest. As a corollary, we obtain when *Q* = *R*, every submodule is honest if and only if every submodule is isolated if and only if *R* is a skew field. We also determine a new characterization of a right Ore domain.

## Publication Title

Applied Categorical Structures

## Volume

8

## Issue

1-2

## First Page

317

## Last Page

326

## Recommended Citation

Fay, T. H.,
Joubert, S. V.
(2000). Isolated Submodules and Skew Fields. *Applied Categorical Structures, 8*(1-2), 317-326.

Available at: https://aquila.usm.edu/fac_pubs/4202