# On completely prime submodules

## TL;DR

This paper advances the study of completely prime modules by exploring their properties, advantages over prime modules, and conditions for fully completely prime modules, including torsion theories related to the completely prime radical.

## Contribution

It extends the theory of completely prime modules, highlighting their advantages, characterizing fully completely prime modules, and introducing torsion theories induced by the completely prime radical.

## Key findings

- Completely prime modules have advantages over prime modules.
- Conditions for fully completely prime modules are characterized.
- Two torsion theories induced by the completely prime radical are presented.

## Abstract

The formal study of completely prime modules was initiated by N. J. Groenewald and the current author in the paper; Completely prime submodules, {\it Int. Elect. J. Algebra}, {\bf 13}, (2013), 1--14. In this paper, the study of completely prime modules is continued. Firstly, the advantage completely prime modules have over prime modules is highlited and different situations that lead to completely prime modules given. Later, emphasis is put on fully completely prime modules, (i.e., modules whose all submodules are completely prime). For a fully completely prime left $R$-module $M$, if $a, b\in R$ and $m\in M$, then $abm=bam$, $am=a^km$ for all positive integers $k$, and either $am=abm$ or $bm=abm$. In the last section, two different torsion theories induced by the completely prime radical are given.