Proving that $\mathbb{C}\otimes_\mathbb{R}\mathbb{C}\cong \mathbb{C}\times \mathbb{C}$ 
Possible Duplicate:
Tensor product $\mathbf{C}\otimes_\mathbf{R} \mathbf{C}$ 

I have to prove $\mathbb{C}\otimes_\mathbb{R}\mathbb{C}\cong \mathbb{C}\times \mathbb{C}$.
By definition of the tensor product, I want to find a map $f:\mathbb{C}\times \mathbb{C}\rightarrow\mathbb{C}\times \mathbb{C}$ such that: 
1) $f(a+a',b)=f(a,b)+f(a',b)$.
2) $f(a,b+b')=f(a,b)+f(a,b')$. 
3) $f(ar,b)=f(a,rb)$.
With the universal property that if $T$ is an abelian group such that there is a mapping $g:\mathbb{C}\times \mathbb{C}\rightarrow T$ that satisfies $1,2,$ and $3$, then $g$ factors through  $f$. 
I was thinking about how to define $f:\mathbb{C}\times \mathbb{C}\rightarrow \mathbb{C}\times \mathbb{C}$ such that property three holds. The problem that I am having is how to find a function such that $f(ar,b)=f(a,rb)$. Could someone maybe help me how to define $f$ and then I could try to go from there and prove that it works. 
Thanks
 A: We can use the algebra of modules:
$$\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}
\cong \mathbb{C} \otimes_{\mathbb{R}} ( \mathbb{R} \oplus \mathbb{R} )
\cong (\mathbb{C} \otimes_{\mathbb{R}} \mathbb{R}) \oplus (\mathbb{C} \otimes_{\mathbb{R}} \mathbb{R})
\cong \mathbb{C} \oplus \mathbb{C}$$
An explicit isomorphism can be obtained by following the diagram. Taking real and imaginary parts for isomorphism $\mathbb{C} \mapsto \mathbb{R} \oplus \mathbb{R}$ (and using the canonical choice for the other isomorphisms) leads to
$$ z \otimes_\mathbb{R} w \mapsto z \otimes_\mathbb{R} (\Re{w}, \Im{w}) \mapsto (z \otimes_\mathbb{R} \Re w, z \otimes_\mathbb{R} \Im w) \mapsto (z \cdot \Re w, z \cdot \Im w)$$
($\Re$ and $\Im$ are for real part and imaginary part)
A: What about $f(a+bi,z)=(az,bz)$?
A: Any two finite dimensional vector spaces over the same field of the same dimension are isomorphic. Since you are tensoring over $\Bbb{R}$ the left hand side has real dimension $2 \times 2$, will the right hand side has real dimension $2 + 2 $. Since these are both $4$ they are isomorphic as real vector spaces.
