Rings with Zassenhaus families of ideals

Let R be a ring with identity. We call a family F of left ideals of R a Zassenhaus family if the only additive endomorphisms of R that leave all members of F invariant are the left multiplications by elements of R. Moreover, if R is torsion-free and there is some left R-module M such that R subset of M subset of R circle times(Z)Q and End(Z)(M) = R we call R a "Zassenhaus ring". It is well known that all Zassenhaus rings have Zassenhaus families. We will give examples to show that the converse does not hold even for torsion-free rings of finite rank.