- Cotorsion modules

A module M is called cotorsion if for all flat modules X, $Ext_R^1(X,M)=0$ .

- Strongly cotorsion modules

M is called strongly cotorsion if for all modules X of finite flat dimension, $Ext_R^1(X,M)=0$ .

- Questions

By definitions it is easy to observe that strongly cotorsion modules are cotorsion modules. But I fail to find some examples to show the two classes are different. Furthermore, what happens when the Ext functor is replace by the Tor functor?

intoa flat module until I found that reference in Weibel. Knowing $X$ has finite flat dimension doesn't tell me $Tor^1(X,M)=0$. I'll bet one can get away with a lesser hypothesis on $M$ than flat in order to conclude this. – David White Jul 30 '12 at 2:19