$k[x]$-module is $k$-vector space

Question

Could anyone tell me why an $A$-module is a $k$-vector space with a linear transformation? (Here $A=k[x]$ where $k$ is a field.)

Thanks!

Answer 1

What is a structure of a module over $k[x]$ on a vector space $V$? It is by definition a map of algebras $k[x]\to End(V)$. How do you describe such maps? Can you see where $1\in k[x]$ must go? Where $x$ can go? If you know this data, do you know where any monomial $x^r$ must go?

Answer 2

Let's unwrap the definitions. An $A$-module $M$ is an abelian group with an action of $A$ satisfying certain axioms. Since $k$ is a subring of $A$, $k$ also acts on $M$, that is, $M$ is a $k$-vector space. The rest of the action is determined by that of $x$ and the module axioms, and since $x$ doesn't satisfy any relations in $A$, we can let $x$ act as any $k$-linear endomorphism of $M$.