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Tensor Product of C with itself over R.

  1. Mar 1, 2013 #1
    I am trying to prove that C[itex]\otimes[/itex]C (taken over R) is equal to C^2. The method I have seen is to show the following equivalences:
    C[itex]\otimes[/itex]C = C[itex]\otimes[/itex](R[T]/<T^2+1>) = C[T]/<T^2+1> = C.
    (All tensor products taken over R).

    The only part I am having trouble with is showing that C[itex]\otimes[/itex](R[T]/<T^2+1>) = C[T]/<T^2+1>. I have tried to show this by using the universal property of the tensor product. First I defined a bilinear map from CXR[T]/<T^2+1> to C[T]/<T^2+1> by sending an element (z,f(T)) to zf(t) (where I am omitting the cosets). This then induces a linear map from the tensor product. However, I cannot seem to find an inverse for this map.

    Any help would be greatly appreciated.
  2. jcsd
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