So, I am trying to imagine what a Module and R-module look like, and what kind of elements are in these structures.
For a Module, it's analogous to a ring of a vector space over a field. So can I think of a Module as a set of vectors? And the entries of the vectors are in the field? What is the general way I think of a module?
For an $R-module$, let $R$ be a ring. An $R-module$ $V$ is an abelian group with addition and scalar multiplication. This means, we have a set of vectors that can be scaled by elements in $R$, and are closed under addition?
This is an example of the module I was reading about, but I don't think I'm imagining the structure correctly. If someone can clarify, that would be great.
$R = C[x,y]$ and $M$ is the ideal of $R$ generated by two elements $x$ and $y$.
Initially I thought, since a module is a set of vectors, M would be some vectors of length 1, because an ideal is a set containing polynomials. But that seems to be incorrect.