What is the rank of $\mathbb{Z}[i] \otimes_{\mathbb{Z}[i]} \mathbb{Z}[i]$ as $\mathbb{Z}$-module?

Question

What is the rank of $P=\mathbb{Z}[i] \otimes_{\mathbb{Z}[i]} \mathbb{Z}[i]$ as $\mathbb{Z}$-module ? Also I have to provide a basis.

My guess is the following. As a $\mathbb{Z}[i]$ module clearly $P \cong \mathbb{Z}[i]$. Since $\mathbb{Z}[i]$ is a $\mathbb{Z}$ algebra, $P \cong \mathbb{Z}[i]$ as $\mathbb{Z}$-module as well.Thus $\text{rank}_{\mathbb{Z}}(P)=2.$ Is it correct ?

How can I give a basis ?

Answer 1

Yes, $P\cong\mathbb{Z}[i]$ as $\mathbb{Z}[i]$-modules. So that we only need to see $\mathbb{Z}[i]$ as a $\mathbb{Z}[i]$-module. Also, $\mathbb{Z}[i]$ is a free $\mathbb{Z}$-module with basis $\{1,i\}$, since $\mathbb{Z}[i]=\{a+ib\colon\ a,b\in\mathbb{Z}\}\cong\mathbb{Z}^2$ (as $\mathbb{Z}$-modules).

Your argument is incomplete unless you specify a basis consisting of two elements or an $\mathbb{Z}$-module isomorphisim from $P$ to $\mathbb{Z}^2$ like above.