Are valuation rings regular? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T06:24:25Z http://mathoverflow.net/feeds/question/37757 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37757/are-valuation-rings-regular Are valuation rings regular? Makhalan Duff 2010-09-04T22:37:16Z 2010-09-04T22:47:53Z <p>This question is short, and to the point:</p> <p>Valuation rings are certainly integrally closed, but are they regular?</p> <p>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.</p> http://mathoverflow.net/questions/37757/are-valuation-rings-regular/37758#37758 Answer by Richard Borcherds for Are valuation rings regular? Richard Borcherds 2010-09-04T22:47:53Z 2010-09-04T22:47:53Z <p>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.) </p>