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2 added 558 characters in body

I think the main reason is the flexibility of working in the category of $R$-modules rather than just with the ring $R$. For instance suppose we stick to rings - we have some ways of building new rings like localization and taking factor rings and limited ways of "building new things" - basically linear combinations of elements and maybe taking limits if $R$ is complete with respect to some topology.

At the level of module categories we still get all of this, torsion theories deal with localization (and make it clear that this is really an "internal" concept), instead of a quotient map we get useful adjoints, and we can still add and compose endomorphisms of $R$. But we also have lots of other structure to work with. We have all limits and colimits, possibly a tensor product, injective modules (which can have a lovely structure theory), duality, etc... So not only can we build a lot more objects but one can prove that just the existence of certain objects gives us a lot of information about the ring.

In a sense (which one can make precise) the category of $R$-modules is the same as $R$ (which is the same as $D^{\mathrm{perf}}(R)$ with its tensor product for $R$ commutative with unit) so the distinction between a ring and its modules shouldn't really exist!

I thought I would add the following quote which sprang to mind when I read the question:

"Grothendieck would later describe each sheaf on a space T as a meter stick measuring T."

taken from McClarty's article The Rising Sea: Grothendieck on simplicity and generality I (which can be found here).

In the commutative with unit case this can be interpreted literally, however, I still think it is relevant (suitably adjusted) in the case of noncommutative rings.

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I think the main reason is the flexibility of working in the category of $R$-modules rather than just with the ring $R$. For instance suppose we stick to rings - we have some ways of building new rings like localization and taking factor rings and limited ways of "building new things" - basically linear combinations of elements and maybe taking limits if $R$ is complete with respect to some topology.

At the level of module categories we still get all of this, torsion theories deal with localization (and make it clear that this is really an "internal" concept), instead of a quotient map we get useful adjoints, and we can still add and compose endomorphisms of $R$. But we also have lots of other structure to work with. We have all limits and colimits, possibly a tensor product, injective modules (which can have a lovely structure theory), duality, etc... So not only can we build a lot more objects but one can prove that just the existence of certain objects gives us a lot of information about the ring.

In a sense (which one can make precise) the category of $R$-modules is the same as $R$ (which is the same as $D^{\mathrm{perf}}(R)$ with its tensor product for $R$ commutative with unit) so the distinction between a ring and its modules shouldn't really exist!