Modules over $k[x]$

Question

Given a ring $R = k[x,y]$, what is the difference between $R$ as a $R[x]$ - module and $R$ as a $k[x]$ module. I just can't seem to comprehend the difference so any help would be appreciated.

Answer 1

Well, what is the difference between $k[x]$ as a $k$-module (vector space) and as a $k[x]$-module? Answer: in the first case it is (free) of infinite countable dimension with basis $\{1,x,x^2,\ldots\}$, and in the second free of rank one with basis $\{1\}$.

Answer 2

If you think of the elements of a modules as "vectors", then the only difference is where the "scalars" come from.

• If $R$ is an $R$-module (which is the same as an $R[x]$-module, since $x\in R$), then your "scalars" come from $R=k[x,y]$. In particular, this module is finitely generated $$R=\left <1\right >_R$$ that is, every element in $R$ can be written as $r\cdot 1$, with $r\in R$
• Likewise, if $R$ is an $k[x]$-module, your scalars come from $k[x]$. In this case $R$ is not finitely generated, since $$R=\left < 1,y,y^2,\dots\right >_{k[x]}$$ that is, every element of $R$ can be written as either $r\cdot 1$, $r\cdot y$, $r\cdot y^2$, and so on, where $r\in k[x]$