Group ring of $\mathbb Z\times \mathbb Z$.

Question

We know that the group ring $\mathbb Z[\mathbb Z]$ of $\mathbb Z$ is just the Laurent polynomial ring $\mathbb Z[t^{\pm1}]$. I want to know about the group ring $\mathbb Z[\mathbb Z\times \mathbb Z]$ of the direct product $\mathbb Z\times \mathbb Z$, does it have a similar description? Is it a Laurent polynomial in two variables $s$ and $t$ : $\mathbb Z[s^{\pm 1},t^{\pm 1}]$ ?