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Is it true that $\mathbb{R}\otimes_\mathbb{Z} \mathbb{R}$ (the tensor product of $\mathbb{R}$ and $\mathbb{R}$ over $\mathbb{Z}$) is not isomorphic to $\mathbb{R}$ as a $\mathbb{Z}$-module? Please give proof.

It is easy to prove that the tensor product of $\mathbb{Q}$ and $\mathbb{Q}$ over $\mathbb{Z}$ is isomorphic to $\mathbb{Q}$ as $\mathbb{Z}$-modules.

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