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I have read the phrase "a good way to study a ring R is to study its modules" many times - though I cannot really see why this should be a general phenomenon.

Is this phenomenon an analogue to (or an instance of?) the fact that studying field extensions of $\mathbb{Q}$ can tell us something about $\mathbb{Q}$ itself? For example, by studying factorizations in $\mathbb{Z}[i]$, one can draw conclusions about factorizations in $\mathbb{Z}$.

Another example is factor rings (If $R$ is a ring and $I$ is in ideal, then $R/I$ as an $R$-module). Yet another example is localizations. For example, if $Y$ is a variety (or a manifold), then the local ring at a point P tells us how functions on $Y$ behave near $P$.

But all these examples seem more like lucky coincidences (because so many objects are R-modules), than an instance of a general phenomenon (that "a good way to study a ring $R$ is to study its modules").

So is there more to the phrase "a good way to study a ring $R$ is to study its modules", or is it just a shorthand way of saying "to study a ring $R$, study its quotients, its localizations and its extension..."?

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Morally: a good way to study an object is by seeing what it can do, and what can be done to it. – user21725 Mar 24 '12 at 22:44
'Groups, as men, will be known by their actions.' -- Guillermo Moreno – Math137 Feb 10 '15 at 11:32
up vote 20 down vote accepted

Here's what always makes sense to me.

Yoneda's lemma tells us that if we really want to understand a ring $R$ we should study the sets $\text{Hom}_{\text{Ring}}(R,S)$ for all the other rings $S$ we could possibly imagine. That said, studying ALL the rings $S$ seems a little naive--is there no way to lighten our load? Well, intuitively if we have some class of rings $\mathscr{S}$ for which every ring $S$ embeds into one of the rings in $\mathscr{S}$ then we should be able to glean all of the information about $R$ by studying $\left\{\text{Hom}_{\mathbf{Ring}}(R,S)\right\}_{S\in\mathscr{S}}$. Now, it's a common fact from basic ring theory that every ring $S$ embeds into the endomorphism ring $\text{End}_{\text{Ab}}(A)$ of some abelian group $A$ (in fact, it's own underlying abelian group structure $(S,+)$) and thus we see that (throwing our indexing cares to the wind) that a candidate for $\mathscr{S}$ is $\left\{\text{End}_\mathbf{Ab}(A)\right\}_{A\in\text{Ab}}$.

But, what is studying an element of $\text{Hom}_\mathbf{Ring}(R,\text{End}_\mathbf{Ab}(A))$ but studying an $R$-module structure on $A$. Thus, to study $\left\{\text{Hom}_\mathbf{Ring}(R,\text{End}_\mathbf{Ab}(A))\right\}_{A\in\text{Ab}}$ is nothing more than to study all the $R$-modules that $R$ admits. And, as we have already said, Yoneda's lemma tells us that intuitively this should comprise all the ring theoretic information about $R$ you could possibly want.

Concretely, notions like Von Neumann regular and semisimple about rings have reformulations about the class of modules over those rings. For example, a ring $R$ is Von Neumann regular if and only if all its left $R$-modules are flat.

Anyways, this was all just what makes sense to me, and so may contains some serious inaccuracies. I hope it helps!

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This thread at MathOverflow discussed this thoroughly:

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