On regular fusible modules

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 not equal m is an element of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\ne m\in M$$\end{document} is said to be regular fusible if there exist r is an element of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in R$$\end{document}, 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 fractions. Also, we characterize division rings in terms of fusible modules.