Suppose that $A=\mathbb{R}[x]$ is the polynomial ring of one variable over the real numbers and $B$ is the real vector space which is the set $B = \{nx\mid n\text{ is a real number}\}$.
Show that $A\otimes_{\mathbb{R}} B \otimes_{\mathbb{R}} A$ is a free $A \otimes_{\mathbb{R}} A$-module.