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..."?