Atypical definition of "regular ring"?
By James Williams
In my adventures I've finally come by a copy of Examples of Commutative Rings by Hutchins, and almost immediately read something that surprised me.
On page 10, a regular ring is defined as Noetherian ring whose every localization at a maximal ideal is a regular local ring.
Up until now, I thought the standard definition was that it was rather a Noetherian ring whose every localization at a prime ideal is a regular local ring.
I'm not handy with commutative algebra... is the definition using maximal ideals a thing? Or are they equivalent and I just haven't run across the theorem?
Am I apparently going to have to re-interpret everything he says about regular rings?
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$\begingroup$The definitions are equivalent. The key fact is that the localization of a regular local ring at a prime ideal is still regular (this follows immediately from the characterization of regular local rings as local rings of finite global dimension, for instance). So, suppose you have a ring $R$ which is regular by Hutchins's definition, and let $P$ be any prime ideal in $R$. Let $M$ be a maximal ideal containing $P$; then $R_M$ is regular local. But $R_P$ is just the localization of $R_M$ at the prime $P_M$, so $R_P$ is also regular local.
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