# Regular ring of infinite dimension?

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, $\dim A=\operatorname{gl}\dim A$.

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For the second question, the answer is also yes because $\dim A$ is the supremum of the dimensions of the localizations at maximal ideals $A_m$, and the same is true for the global dimensions (Matsumura, Commutative Algebra, p. 130, Lemma 5.III.)
Edit I didn't see the first part of the second question. So the equivalence is true if $A$ has finite dimension.