  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.
