The Natural partial order on modules

TL;DR

This paper explores the Mitsch order on modules, establishing its properties, relationships with other orders, and demonstrating that it forms a partial order with specific lattice characteristics.

Contribution

It introduces a module-theoretic version of the Mitsch order, proves it is a partial order, and analyzes its lattice properties and relations to other known module orders.

Findings

Mitsch order is a partial order on modules.

The minus order coincides with the Mitsch order.

Counterexample shows the converse of the minus order being the Mitsch order does not hold.

Abstract

The Mitsch order is already known as a natural partial order for semigroups and rings. The purpose of this paper is to further study of the Mitsch order on modules by investigating basic properties via endomorphism rings. And so this study also contribute to the results related to the orders on rings. As a module theoretic analog of the Mitsch order, we show that this order is a partial order on arbitrary modules. Among others, lattice properties of the Mitsch order and the relations between the Mitsch order and the other well-known orders, such as, the minus order, the Jones order, the direct sum order and the space pre-order on modules are studied. In particular, we prove that the minus order is the Mitsch order and we supply an example to show that the converse does not hold in general.