I've taken an undergraduate course in ring theory, but in my graduate algebra course we just started modules, and I'm kind of confused.

- What's the difference between modules and Ideals?

- Are modules not closed under multiplication?

- What criteria is necessary to show something is a submodule? In my definition I just have, "closed under the action of ring elements (rn in N for a r in R n in N). But I'm not quite sure what that means. In some examples they just show a + rb is in N for all r in R, a,b in N. But sometimes they show r1(a+r2b) in N. What do I have to show in general?

In the book they keep talking about how modules are very similar to vector spaces... but I'm not familar with the definition of a vector space, so could someone please explain it just in relation to rings and Ideal? Thanks.