Just started studying tensor products. Let $A$ be a commutative ring with unity and let $M$ be an $A$-module. Now let $k$ be a field, I know that a $k$-module is precisely a $k$-vector space.
My question is the following:
Why $k \otimes_{A} M$ is also a $k$-vector space? here $\otimes$ denotes the tensor product with respect the ring $A$.
Is it because we can give $k \otimes_{A} M$ the structure of $k$-module by just taking:
$f: k \times (k \otimes_{A} M) \rightarrow k \otimes_{A}M$ given by:
$f(c,d \otimes m)=(cd) \otimes m$ ? where $c,d \in k$