Hi there, just trying to wrap my head around the notion of a module, and I'm wondering could someone help me? I'm tying in information I've gleaned from a few disparate modules in college, so I may be wrong about a few things, feel free to correct me!

Ok so I understand that they are a generalisation of a vector space, where the corresponding scalars lie in a ring rather than a field, but I'm unsure as to the limitations or differences this causes?

Also, I know you can represent a finite vector from a vector space over a field in the form $\displaystyle v = (a_{1}, a_{2}, ..., a_{n})$. Is there a way to represent the elements of a module similarly? Are those elements even similar to vectors? Or must I just think of it as a purely abstract object which is affected by the operations on some ring R?

Thanks!