On q-commutative power and Laurent series rings at roots of unity

We continue the first and second authors? study of q-commutative power series rings and Laurent series rings , 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 results establish 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.