# Minus Partial Order in Regular Modules

## TL;DR

This paper extends the minus partial order concept to regular modules, providing new characterizations and generalizations, thereby broadening its applicability beyond matrices and operators.

## Contribution

It introduces the minus partial order in the context of regular modules and proves it forms a partial order, with multiple characterizations and generalizations.

## Key findings

- Minus partial order is a partial order in regular modules.
- Provides new characterizations of the minus partial order.
- Generalizes known results to the module setting.

## Abstract

The minus partial order is already known for sets of matrices over a field and bounded linear operators on arbitrary Hilbert spaces. Recently, this partial order has been studied on Rickart rings. In this paper, we extend the concept of the minus relation to the module theoretic setting and prove that this relation is a partial order when the module is regular. Moreover, various characterizations of the minus partial order in regular modules are presented and some well-known results are also generalized.