* 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?



Do the classes of cotorsion and strongly cotorsion modules coincide?
Let me raise an example:


Let $R = k[[t]]$, $k$ be a field. $R$ is a regular ring as well as a Dedekind ring. 


1. $R$ is regular hence every $R$-module $X$ has finite flat dimension. Thus if $M$ is strong cotorsion, then $Ext^1\_R(X,M) = 0$ for all $X$, so $M$ is injective.
2. By (1) we have $k$ is not strong cotorsion. We shall show that $k$ is cotorsion.
3. Since $R$ is Dedekind we have $X$ is flat iff $X$ is torsion-free (see <http://en.wikipedia.org/wiki/Flat_module>). So we have a short exact sequence
$$0 \to X \to X \to X/tX \to 0.$$
The above exact sequence induces the following exact sequence
$$Ext^1\_R(X,k) \overset{t}{\to} Ext^1\_R(X,k) \to Ext^2\_R(X/tX,k) \cong 0$$
the isomorphism $Ext^2\_R(X/tX,k) \cong 0$ follows from $R$ is a regular ring of dimension one.
But $Ext^1\_R(X,k) \overset{t}{\to} Ext^1\_R(X,k)$ is a zero homomorphism. So $Ext^1\_R(X,k) \cong 0$. The proof is complete


