  We continue the first and second authors' study of $q$-commutative power
series rings $R=k_q[[x_1,\ldots,x_n]]$ and Laurent series rings $L=k_q[[x^{\pm
1}_1,\ldots,x^{\pm 1}_n]]$, specializing to the case in which the commutation
parameters $q_{ij}$ are all roots of unity. In this setting, $R$ is a PI
algebra, and we can apply results of De Concini, Kac, and Procesi to show that
$L$ is an Azumaya algebra whose degree can be inferred from the $q_{ij}$. Our
main result establishes an exact criterion (dependent on the $q_{ij}$) for
determining when the centers of $L$ and $R$ are commutative Laurent series and
commutative power series rings, respectively. In the event this criterion is
satisfied, it follows that $L$ is a unique factorization ring in the sense of
Chatters and Jordan, and it further follows, by results of Dumas, Launois,
Lenagan, and Rigal, that $R$ is a unique factorization ring. We thus produce
new examples of complete, local, noetherian, noncommutative, unique
factorization rings (that are PI domains).
