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Hi

I know how to calculate some easy tensor products like $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}\cong_{\mathbb{Z}} \mathbb{Z}/(m,n)\mathbb{Z} $ or $F[X] \otimes_{F} F[Y] \cong_F F[X,Y]$ or $\mathbb{C} \otimes_\mathbb{R} \mathbb{C} \cong_\mathbb{R} \mathbb{R}^4$ but the reason that I can do this is that the structure of these rings is kinda well-known or they're finitely generated modules I know that continuous real functions over the interval $[0,1]$ form an $\mathbb{R}$-module, so, as a challenge for myself, I tried to find $C([0,1]) \otimes_\mathbb{R} C([0,1])$, but nothing came to my mind. So I thought I would need help and hence that's why I'm writing it here now. I also like to know what $\mathbb{R} \otimes_\mathbb{Q} \mathbb{R}$ looks like, is it algebraically the same as $\mathbb{R}$?

Thanks in advance.

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    $\begingroup$ The tensor product ${\mathbf R}\otimes_{\mathbf Q}{\mathbf R}$ is isomorphic to ${\mathbf R}$ as a real vector space since the dimensions are the same, but it's not isomorphic to $\mathbf R$ as a ring since it contains zero divisors. The tensors $\sqrt{2} \otimes 1$ and $1 \otimes \sqrt{2}$ are linearly independent over $\mathbf Q$ and square to $2(1 \otimes 1)$, so $(\sqrt{2} \otimes 1 + 1 \otimes \sqrt{2})(\sqrt{2} \otimes 1 - 1 \otimes \sqrt{2}) = 2 \otimes 1 - 1 \otimes 2 = 0$ with neither factor on the left side being 0. $\endgroup$
    – KConrad
    Mar 7, 2013 at 14:12
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    $\begingroup$ Please clarify if you are interested in the structure of these tensor products as $R$-modules or as $R$-algebras (with $R$ being the ring you are tensoring over). $\endgroup$
    – KConrad
    Mar 7, 2013 at 14:15
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    $\begingroup$ By the way, I'm new here, I didn't know that here is filled with PhD students and professors that ask really sophisticated questions. I'm just a 2nd-year undergraduate student, for me, tensors, is the highest level of mathematics that I've ever encountered! So, please, go easy on me and explain things to me in the simplest way possible. $\endgroup$ Mar 7, 2013 at 15:15
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    $\begingroup$ @KConrad: How is ${\mathbb R}\otimes_{\mathbb Q}{\mathbb R}$ as a real vector space isomorphic to $\mathbb R$? To start with, how is the vector space structure defined? One can multiply on the first or the second slot. Let's say the first slot, then $\sqrt 2\otimes 1$ and $1\otimes 1$ are linearly dependent over $\mathbb R$, but $1\otimes \sqrt 2$ and $1\otimes 1$ are not. It rather seems to me that ${\mathbb R}\otimes_{\mathbb Q}\mathbb R$ has the same diemnsion over $\mathbb R$ as $\mathbb R$ has over $\mathbb Q$? $\endgroup$
    – user1688
    Mar 7, 2013 at 16:05
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    $\begingroup$ @some1.new4u: Then perhaps math.stackexchange.com is the more appropriate site for you to ask your questions. MO is for research-level questions. $\endgroup$ Mar 7, 2013 at 18:16

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