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'''Example:''' We propose to compare <math>\R \otimes_{\Z} \R </math> and <math>\R \otimes_{\R } \R </math>. Like in the previous example, we have: <math>\R \otimes_{\Z} \R = \R \otimes_{\Q} \R </math> as abelian group and thus as <math>\Q</math>-vector space (any <math>\Z</math>-linear map between <math>\Q</math>-vector spaces is <math>\Q</math>-linear). As <math>\Q</math>-vector space, <math>\R </math> has dimension (cardinality of a basis) of [[Cardinality of the continuum|continuum]]. Hence, <math>\R \otimes_{\Q } \R </math> has a <math>\Q</math>-basis indexed by a product of continuums; thus its <math>\Q</math>-dimension is continuum. Hence, for dimension reason, there is a non-canonical isomorphism of <math>\Q</math>-vector spaces:
<math display="block">\R \otimes_{\Z } \R \approx \R \otimes_{\R } \R .</math>
<!-- but they are not isomorphic as rings since the ring on the left is not even a field. -->
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