On the derived quotient module

With every R-module M associate the direct limit of ${\operatorname{Hom}_R}(D,M)$ over the dense right ideals of R, the derived quotient module $\mathcal{D}(M)$ of M. $\mathcal{D}(M)$ is a module over the complete ring of right quotients of R. Relationships between $\mathcal{D}(M)$ and the torsion theory of Gentile-Jans are explored and functorial properties of $\mathcal{D}$ are discussed. When M is torsion free, results are given concerning rational closure, rational completion, and injectivity.