So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only examples I'm really comfortable with are


* vector spaces,
* finitely generated modules over a PID, and
* modules over a group algebra.


But when I try to apply these examples to understanding something like Nakayama's lemma I don't have any intuition to bring to the table. So, what other examples of modules should I keep in mind so that


* I'm not fooled by my intuition about vector spaces, and
* I can concretely understand what something like Nakayama's lemma means, at least in an important special case?



What representative examples of modules should I keep in mind?
Yes, there is a big class of modules that have an intuition different from the abstract algebra, namely the ones that come from an **algebraic geometry**. If $R$ is a (say, Noetherian) commutative ring, then you consider a scheme $\mathrm{Spec}\, R$ and (finitely generated) modules over $R$ correspond to coherent sheaves on $\mathrm{Spec}\, R$.


For example, a **skyscraper sheaf** on point $p$ corresponds to the module $R/m\_p$, where $ m\_p$ is a maximal ideal that defines point. More interestingly, a **locally trivial bundle** over $\mathrm{Spec}\, R$ corresponds to a projective module (this is exactly the statement locally free = projective).


The objects that seem unnatural on algebraic side, like support of a module, in fact have a clear geometric meaning — **support** of a sheaf is where the sheaf sits, that's it. And this is the geometric setting that explains Nakayama's lemma: the lemma says you can consider things locally for a coherent sheaf (see [wikipedia](http://en.wikipedia.org/wiki/Coherent_sheaf#Definition)).


