I have some exercises on tensor products of modules and algebras and I'm a little bit confused. This is our definiton of a tensor product:
If $X$ is a right $R$-module and $Y$ is a left $R$-module, the tensor product $X \otimes _R Y$ is defined to be the additive group generated by symbols $x \otimes y \,\,(x \in X, y \in Y )$, subject to the relations ($x,x_0 \in X, y,y_0 \in Y)$:
- $(x + x_0 ) \otimes y = x \otimes y + x_0 \otimes y$,
- $x \otimes (y + y_0 ) = x \otimes y + x \otimes y_0$,
- $(xr) \otimes y = x \otimes (ry)$ for $r \in R$.
If we now have a field $K$ and two $K$-vector spaces $X,Y$, this yields the tensor product of vector spaces we know (like it is defined on Wikipedia for example), right?
On vector spaces we can always choose a basis and then the tensor product is already determined on the whole space $X \otimes Y$ by a definition on it's basis.
Now I need to show that $\mathbb{C} \otimes_\mathbb{R} \mathbb{C} \cong \mathbb{C} \times \mathbb{C}$ as $\mathbb{R}$-algebras. My idea is to choose a $\mathbb{R}$-basis of $\mathbb{C}$ and then just define an isomorphism on this basis. But I guess that's wrong since a hint of this exercise says that I should use the Chinese Remainder Theorem and that's really confusing me. First, what has the Chinese Remainder Theorem to do with this exercise? Second, what's wrong with my idea? Why is it not that easy? Is it because we are dealing with algebras and not only with modules?
And also I have another exercise where I have the following task: Regard $\mathbb{H}$ as a left $\mathbb{C}$-module with basis $\{1, j\}$, and we have an injective $\mathbb{R}$-algebra homomorphism $\mathbb{H} \rightarrow M_2(\mathbb{C}), z + wj \mapsto \left(\begin{matrix} z & w \\ -\overline{w} & \overline{z} \\ \end{matrix}\right)$.
Show that this induces an isomorphism of $\mathbb{C}$-algebras $\mathbb{H} \otimes_\mathbb{R} \mathbb{C} \rightarrow M_2(\mathbb{C})$.
Here my question is, the elements of $\mathbb{H} \otimes_\mathbb{R} \mathbb{C}$ are of the form $h_1 \otimes 1 + h_2 \otimes i$, are they? And if so, can I just directly extend the homomorphism to $\mathbb{C}$-algebras $\mathbb{H} \otimes_\mathbb{R} \mathbb{C}$?