I am having a little difficulty in understanding the module structure of tensor products of modules over algebras (as opposed to arbitrary rings). To help me understand, here is an example.
Let $K[x]$ be the polynomial algebra in one variable over an algebraically closed field $K$. Let $A$ be a $K$-algebra. Suppose $M$ is a $K[x]$-$A$-bimodule such that $M$ is finitely generated and free as a left $K[x]$-module. (By free, we mean that $M$ is isomorphic to a direct sum of copies of $K[x]$, where the algebra is considered as a left module over itself.) Let $S$ be a simple right $K[x]$-module.
Consider the module $S \otimes_{K[x]} M$. This is a right $A$-module.
How does one determine the underlying K-vectorspace of $S \otimes_{K[x]} M$? Supposedly, this module is finite-dimensional, although this confuses me, since I thought the dimension was multiplicative over a tensor product (and $K[x]$ is an infinite-dimensional algebra). Why?
Finally, what is the right action of an element $a \in A$ on $S \otimes_{K[x]} M$?