Is it possible for a (non-local) regular ring to have an infinite dimension? As far as I know, the well known characterization of regular rings in terms of finite global dimension is only for local case. Or does it also apply to non-local regular rings? Specifically, is the following statement true?
A (non-local) Noetherian ring $A$ is regular if and only if it has a finite global dimension. In this case, $\dim A=\operatorname{gl}\dim A$.