TITLE: Can we really understand $R$ by studying $R$-modules?
QUESTION [7 upvotes]: According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. 
Can someone give some honest examples where the category of modules do tell us a lot about the underlying ring? Better still, such kind of information is very difficult if one looks at the ring directly.
Thanks very much!

REPLY [6 votes]: Regarding the study of structure of rings using internal structure such as ideals vs. external structure such as general modules over the ring, you might find of interest the following nice introduction in J.P. Lafon: Ideals and Modules, in Hazewinkel (ed.), Handbook of Algebra, vol.1.