Simple tensors in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{R}$ My notes state that every simple tensor of $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{R}$ can be written as $1 \otimes x + i \otimes y$. 
I can't quite convince myself of this statement. 
Every element of $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{R}$ is a finite linear combination over $\mathbb{Z}$ of simple tensors $z \otimes x$ where $z \in \mathbb{Z}[i]$ and $x \in \mathbb{R}$. Letting $z=a+bi$, we have that 
$$(a+bi) \otimes x = a \otimes x + b i \otimes x.$$ 
This is not quite the property above though. 
How do we arrive at the above form for a simple tensor?  

Notice that for an element $z\in\Bbb Z[i]$ comes in the form $a+bi$ with $a,b\in\Bbb Z$, as such we have for any tensor that
$$\sum_j (a_j+b_ji)\otimes_\Bbb z r_j = \sum_j a_j \otimes_\Bbb z r_j + \sum_j b_ji \otimes_\Bbb z r_j$$
$$=\sum_j  1\otimes_\Bbb z a_jr_j + \sum_j i \otimes_\Bbb z b_jr_j$$
$$=  1\otimes_\Bbb z \sum_j a_jr_j + i \otimes_\Bbb z \sum_jb_jr_j$$
$$=  1\otimes_\Bbb z x + i \otimes_\Bbb z y$$
Which shows it.
