# A relationship between 2-primal modules and modules that satisfy the radical formula

## TL;DR

This paper explores the connection between 2-primal modules and modules satisfying the radical formula, highlighting their role in understanding modules over both commutative and noncommutative rings, with new examples provided.

## Contribution

It introduces the concept of 2-primal modules and compares them with modules satisfying the radical formula, bridging the gap between commutative and noncommutative module theory.

## Key findings

- Identification of 2-primal modules as a key concept
- Examples of rings and modules satisfying the radical formula
- Demonstration of the importance of 2-primal modules in module theory

## Abstract

The coincidence of the set of all nilpotent elements of a ring with its prime radical has a module analogue which occurs when the zero submodule satisfies the radical formula. A ring $R$ is 2-primal if the set of all nilpotent elements of $R$ coincides with its prime radical. This fact motivates our study in this paper, namely, to compare 2-primal submodules and submodules that satisfy the radical formula. A demonstration of the importance of 2-primal modules in bridging the gap between modules over commutative rings and modules over noncommutative rings is done and new examples of rings and modules that satisfy the radical formula are also given.