Torsions in modules

In the category of right modules over the ring $E=\operatorname{(End}_R(F)$, where $F$ is a free right R-module, a torsion is defined. It is known as Tol'skaya torsion. The correlation between torsion-free $E$-modules in the sense of Tol'skaya and torsion-free $E$-modules in the sense of Bass is investigated. It is shown that the ring $R$ is a right cogenerator if and only if in the ring of endomorphisms of any free $R$-module, $r(l(J))$ for all finitely generated right ideals $J$.