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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.

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Please don't post questions in the imperative mode. If you have a question, please ask. What have you attempted, or where are you stuck? –  Arturo Magidin Mar 7 '11 at 1:55

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Since $B\cong \mathbb{R}$ as a vector space/module, what does that tell you about $A\otimes_{\mathbb{R}} B$? And then about $(A\otimes_{\mathbb{R}}B)\otimes_{\mathbb{R}} A$?

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