Integral group ring of the free group on two generators

Question

I know that the integral group ring $\mathbb Z[\mathbb Z]$ of $\mathbb Z$ is described as the ring of Laurent polynomials $\mathbb Z[t^{\pm}]$. I'm asking if there is a known description of the integral group ring $\mathbb Z[F_2]$ of the free group on two generators. Thank you for your help!.

Answer 1

The description is the same as the $\,\mathbb Z[\mathbb Z]\,$ case you gave except the polynomials have non commuting variables. Thus the group ring $\,\mathbb Z[F_2]\,$ is described by $\,\mathbb Z[t^{\pm},u^{\pm}]\,$ where the two variables $\,t\,$ and $\,u\,$ do not commute.