TITLE: Derived functors of torsion functor
QUESTION [12 upvotes]: Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a homomorphism, then $f(M^{tor}) \subseteq N^{tor}$, so call $f^{tor} \colon M^{tor} \to N^{tor}$ the induced map. We have a covariant functor $^{tor}$ from the category of $A$-modules to itself. It is straightforward to verify that $^{tor}$ is an additive left-exact functor; so we can consider its right-derived functors $R^i$, for $i \geq 0$: if $Q^{\bullet}$ is an injective resolution of $M$, then $R^i(M) = H^i((Q^{\bullet})^{tor})$.
If $A$ is a PID then it is quite easy to compute $R^i(M)$ for finite $A$-module $M$, because it is easy to have injective resolutions. In fact it is well-known that $K$ and $K/A$ are injective $A$-modules, where $K$ is the field of fractions of $A$.
My questions are:

Can one compute $R^i(M)$ if $A$ is not a PID or $M$ is not finite? Have these functors been studied?
What is the relation between $R^i$ and $\mathrm{Tor}_j$?
In the category of $A$-modules, are there other derived functors that have been studied and that are not $\mathrm{Tor}$ nor $\mathrm{Ext}$?

REPLY [6 votes]: For 3: This is almost cheating, I know...
Mac Lane studied the functors $\mathrm{Trip}_n$ which arise from derivating (?) the functor $M\otimes N\otimes P$ of three variables. There are references in his book Homology. The interesting thing is, this cannot be expressed in terms of $\mathrm{Tor}$.
For 1: If $\mathcal I$ is the set of all non-zero ideals ordered by inclusion (which is directed), we have $$M^{tors}=\varinjlim_{I\in\mathcal I}\:\hom_A(A/I,M).$$ You should look up the relationship between right-deriving , and directed direct limits like this one, and that should tell you what your $R^i$ functors are. (This will tell you that your $R^i$ are more related to $\mathrm{Ext}$ than to $\mathrm{Tor}$)