# Tensor Product of C with itself over R.

1. Mar 1, 2013

### Monobrow

I am trying to prove that C$\otimes$C (taken over R) is equal to C^2. The method I have seen is to show the following equivalences:
C$\otimes$C = C$\otimes$(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$\otimes$(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.