1. Consider the ring of Laurent polynomials $R := \mathbb{Z}[s,s^{-1}]$ with integer coefficients. Are all projective $R$-modules free? (Let's say left modules by convention.)
2. More generally, let $G$ be the **free** group on a finite set $S$ of generators, and consider the integral group ring $R := \mathbb{Z}G$. Are all projective $R$-modules free? Note that in part (1), $S$ was the singleton $\{ s \}$.
3. Is the answer the same if we allow an infinite set $S$ of generators? Note that the group $G$ itself is infinite either way.


I feel like work of Serre, Swan, Bass, Kaplansky, Suslin, Quillen, and others may be relevant, but I couldn't find the answers.


Thank you for your help.



Projective modules over free groups
The answer to your questions is 'yes' and, as you feel, it is a result of Bass:


MR0178032 (31 #2290) 
Bass, Hyman
Projective modules over free groups are free. 
J. Algebra 1 1964 367–373. 


