An $R$-module $M$ is called torsion if for every $m \in M$, there exists a non-zero divisor $r \in R$ s.t. $rm = 0$. If we let $S$ be the multiplicative system of non-zero divisors of $R$, we get that $M$ is torsion iff
$$ \operatorname{Tor}_1^R(M, R_S/R) = M $$
A module $M$ is called cotorsion if
$$ \operatorname{Ext}^1_R(F, M) = 0$$
for every flat $R$ module $F$.
In what sense are these two definitions dual? I see that one uses Tor while the other uses Ext, but apart from that, there are some asymmetries: one asks that Tor = $M$, while the other asks that Ext = $0$. Furthermore, one looks at $R_S/R$, while the other looks at any flat module $F$. Can the definitions be rephrased such that the duality works out nicely?