Q: Good examples of modules and algebras? I'm going through Atiyah-Macdonald on my own, and I'm feeling a distinct lack of examples. With rings I can just about get by with quotients of polynomial rings and $\mathbb{Q}$ and so forth. Localizations of $\mathbb{Z}_{\mathfrak{p}}$.
With modules though, I'm very unsure of what a good prototypical example would be. $\mathbb{C}[X]$ as a module over $\mathbb{C}$ feels very tame, since this is an infinite dimensional vector space. On the other hand, eg $\mathbb{C}[x,y] / (x^2,y)$ as a module over $\mathbb{C}[x,y]$ feels unweildly, and besides, a quotient seems like a fairly specific kind of module as well. These kinds of arbitrary quotients feel very randomly chosen.
And an algebra, I suppose there's always $\mathbb{C}[X]$ over $\mathbb{C}$ again, but, again this feels like a very easy example.
In topology I feel like I have a strong suite of examples, and even in group theory, there's permutation groups, dihedral groups, matrices, integers mod p additively and multiplicatively, the klein 4 group. I feel I have a reasonably good collection of examples. With modules, I feel much more lost.
What are some good central examples of modules? Which aren't so nice that everything is true of them, but aren't so pathological that they're difficult to think about? And likewise, algebras?
I've seen a duplicate here which wasn't ever answered.
