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Minus Partial Order in Regular Modules

Burcu Ungor, Sait Halicioglu, Abdullah Harmanci, Janko Marovt

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.

Minus Partial Order in Regular Modules

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.

Paper Structure

This paper contains 3 sections, 15 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Minus partial order in modules
  3. Some Characterizations of the Minus order on Modules

Key Result

Proposition 2.4

Let $M$ be a module and $m_1, m_2\in M$. If $m_1\leq^{-} m_2$, then $m_1R\subseteq m_2R$.

Theorems & Definitions (37)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • Definition 2.7
  • ...and 27 more