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This might be a stupid question, but I could not find a good reference that thoroughly explains the matter. I will start with some lengthy introduction.

If $\mathcal{A}$ is any abelian category, then we have functors $\mathrm{Hom}_\mathcal{A} (M,-)$ and $\mathrm{Hom}_\mathcal{A} (-,N)$, both left exact, covariant and contravariant, so if we have enough injectives (resp. projectives), we can calculate their right derived functors $\mathrm{Ext}^n_\mathcal{A} (M,-)$ and $\mathrm{Ext}^n_\mathcal{A} (-,N)$.

Now if $\mathcal{A}$ is the category or $R$-modules, things become more interesting: $\underline{\mathrm{Hom}}_R (-,-)$ has also an $R$-module structure, so it is not merely a functor with values in abelian groups, and $\mathrm{Ext}^n_R (M,N)$ has an $R$-module structure. Of course, the forgetful functor $R\mathcal{-Mod} \to \mathcal{Ab}$ is exact, so if we are not interested in the module structure, we can just derive the $\mathrm{Hom} (-,-)$ with values in $\mathcal{Ab}$. I think many introductory texts or classes ignore the $R$-module structure.

Things start to confuse me when $R$ is non-commutative. We have to distinguish left, right modules, and bimodules. So which $\mathrm{Hom} (-,-)$ functor we should derive (defined on which categories of modules and bimodules and with values in which category of modules/bimodules)? For instance, I think the Yoneda Ext naturally has a bi-module structure (for extensions we take pullbacks or pushouts over the action maps $r\colon M\to M$ and $r\colon N\to N$).

And I have the same question for tensor products and $\mathrm{Tor}$ functors for modules over noncommutative rings.

Edit: I actually wanted to see some reference that, for example, treats $\mathrm{Ext}$ as a derived functor of something like $\underline{\mathrm{Hom}}_S (-,-)\colon (R,S)\text{-bimod}^\mathrm{op} \times (T,S)\text{-bimod} \to (T,R)\text{-bimod}$ but this might be some ridiculous generality.

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    $\begingroup$ There is nothing ridiculous about what you want to do. It is completely standard, and was done in that generality already in the founding text on the subject, the one by Cartan and Eileberg. You could look there, in fact. $\endgroup$ – Mariano Suárez-Álvarez Jul 5 '16 at 0:02
  • $\begingroup$ regarding the Yoneda composition and rhe bimodule structure in terms of pullbacks and pushforwards, you consult the book by Hilton and Stammbach, or MacLane's *Homology*/ $\endgroup$ – Mariano Suárez-Álvarez Jul 5 '16 at 0:03
  • $\begingroup$ (By the way, please do not engage with users who post garbage such as the one in the answer I deleted. Simply flag the post for moderator attention. It is a much better use of your time!) $\endgroup$ – Mariano Suárez-Álvarez Jul 5 '16 at 0:05
  • $\begingroup$ @Mariano Suárez-Alvarez Yes, I learned Yoneda Ext from the MacLane's book, and there I understood that it has a very natural bimodule structure, so one must not forget about the action of R on the usual Exts. Thank you very much for the reference to Cartan-Eileberg! $\endgroup$ – J. Doe Jul 5 '16 at 0:11
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$\text{Hom}_R$ is a functor taking as input two (right, say) $R$-modules and outputting an abelian group. $\otimes_R$ is a functor taking as input a right $R$-module and a left $R$-module and outputting an abelian group. Both of these functors can be derived as usual, and the generalization to bimodules follows from functoriality.

A standard reason to want to do this is to define Hochschild (co)homology, group (co)homology, and Lie algebra (co)homology. I believe these are in fact the classical motivating examples behind the development of homological algebra in general.

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