What is the $k$ action on the Clifford algebra $C(V,q)$?

Question

Let $k$ be a field. I know that Clifford algebra $C(V,q)$ is central simple $k$-algebra where $\dim V$ is $2n$ and $q$ is non degenerate. But what does it mean $k$-algebra here? What is the $k$ action on $C(V,q)$? Is it $a\otimes w$ for $a \in k$ and $w\in C(V,q)$? If we want to prove $C(V,q)$ has central $k$ that is all element of $k$ commute with $ C(V,q)$, should I prove that $1\otimes e_i=e_i \otimes 1 $ for all basis element $e_i$ of $V$?