Let $M$ be an $R$-module for some ring $R$. We usually define the torsion submodule $\mathrm TM\subseteq M$ as $\mathrm TM=\{m\in M\colon \exists r\in R, r\nmid 0\colon rm=0 \}$. We call $M$ torsion if $\mathrm TM=M$.
This is quite different from the definition of a cotorsion module: $M$ is cotorsion if $\operatorname{Ext}^i(F,M)=0$ for all torsion-free modules $F$.
My questions
- does every module have a cotorsion quotient?
- what does not look like, i.e., what is the respective kernel?
- is there a homological definition of a torsion module? Something like $M$ is torsion if $\operatorname{Tor}_1(M, F)$ for every torsion-free (free?) module $F$.
Own thoughts
To adress the first question, one could do the following:
$$\mathrm TM=\varinjlim_{\text{torsion modules $T\subseteq M$}}T\subseteq M.$$
It is a submodule of $M$ by exactness of $\varinjlim$, and it is torsion because any element of $\mathrm TM$ has as representative a torsion element.
Now, we already know what a cotorsion module is. What is
$$M\to\varprojlim_{\text{cotorsion modules $M\twoheadrightarrow C$}} =:\mathrm CM?$$
Is this even a quotient of $M$? Is it cotorsion? If not, does {cotorsion quotient modules of $M$} even have a unique minimal element?