TITLE: Why is it a good idea to study a ring by studying its modules?
QUESTION [89 upvotes]: This is related to another question of mine.  Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module.  You tell him that modules generalize ideals and quotients, but he remains unimpressed.  How do you convince him that studying modules of a ring is a good way to understand that ring?  (In other words, why does one have to work "external" to the ring?)  Your answer should also explain why it is a good idea to study a group by studying its representations.

REPLY [3 votes]: If your friend knows well rings and ideals, he has certainly encountered quotient like $I/J$, hasn't he? What are they for him? sets? abelian groups? This point of view is wanting. 
You can formulate basic and very useful results, such as (for $A$ local, noetherian): if $I/mI$ has a set of $n$ generators, so has the ideal $I$ (Nakayama), only if you can see $I/J$ as a module...