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?