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On Regular Fusible Modules

Osama A. Naji, Mehmet Özen, Ünsal Tekir, Suat Koç

Abstract

In this article, we introduce the notion of regular fusible modules. Let $R$ be a ring with an identity and $M$ an $R$-module. An element $0\neq m\in M$ is said to be regular fusible if there exists $r\in R$, a non zero-divisor of $M$, such that $mr$ can be written as the sum of a torsion element and a torsion free element in $M$. $M$ is called regular fusible if every nonzero element of $M$ is regular fusible. We characterize regular fusible modules in terms of fusible modules. In addition, we show that a regular fusible module over a right duo ring is reduced and nonsingular. Moreover, we study the regular fusible property under Cartesian product, trivial extension ring, and module of a fractions. Also, we characterize division rings in terms of fusible modules.

On Regular Fusible Modules

Abstract

In this article, we introduce the notion of regular fusible modules. Let be a ring with an identity and an -module. An element is said to be regular fusible if there exists , a non zero-divisor of , such that can be written as the sum of a torsion element and a torsion free element in . is called regular fusible if every nonzero element of is regular fusible. We characterize regular fusible modules in terms of fusible modules. In addition, we show that a regular fusible module over a right duo ring is reduced and nonsingular. Moreover, we study the regular fusible property under Cartesian product, trivial extension ring, and module of a fractions. Also, we characterize division rings in terms of fusible modules.
Paper Structure (2 sections, 20 theorems)

This paper contains 2 sections, 20 theorems.

Table of Contents

  1. Introduction
  2. Regular Fusible Modules

Key Result

Proposition 1

If $T_{R}(M)$ forms a submodule of $M$, then $M$ is torsion free if and only if $M$ is regular fusible.

Theorems & Definitions (40)

  • Definition 1
  • Example 1
  • Example 2
  • Example 3
  • Proposition 1
  • proof
  • Corollary 1
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 30 more