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This question is short, and to the point:

Valuation rings are certainly integrally closed, but are they regular?

The motivation is that I'm trying to understand the resolution of singularities of algebraic surfaces as was done originally, and I'm playing around with some of the ideas involved.

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up vote 9 down vote accepted

Regular local rings are Noetherian by definition, but valuation rings are not unless they happen to be discrete valuation rings. So with the usual definitions, most valuation rings are not regular. (It might be possible to come up with a reasonable definition of regularity for non-Noetherian rings, but I have not heard of one.)

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