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I have been trying to find out what the definition of a noncommutative regular local ring is, but to no avail. In fact, how does one even begin to define Krull dimension for a noncommutative ring? Hence, I would appreciate it if someone could kindly provide definitions for the following, in the case when the ring under study is noncommutative:

  • Regular. In the commutative case, the definition of regular involves localizing at prime ideals. However, in the noncommutative case, how do we do localization? Is Ore's Condition invoked somewhere?
  • Regular local. In the commutative case, the definition of regular local involves Krull dimension. However, in the noncommutative case, do we have an analogue of Krull dimension?

On a different note, in the commutative case, is it true that a local ring that is regular the same as a regular local ring? (This might seem to be a stupid question.)

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why not just use the projective dimension definition? – the L Dec 12 '12 at 15:20
Unfortunately, noncommutative algebraists are not so lucky as to have every commutative concept find a noncommutative analogue with all the same features :) Nevertheless, I like the question! – rschwieb Dec 12 '12 at 15:22
up vote 1 down vote accepted

Noncommutative localization is a highly nontrivial concept! There have been practical extensions of localization to noncommutative rings, but the thing to know is that it is not nearly as nice as commutative localization.

For a good survey of noncommutative localization, you can check out all of chapter 9 in T.Y. Lam's Lectures on Modules and Rings.

Another very advanced book on localization ideas is Bo Stenström's Rings of Quotients. I know that Lambek also has a book Noncommutative Localization, but I have not had the chance to read it.

The motivation for studying regular local rings is their geometric connection with regular points. Since I know so little about noncommutative geometry, I can't make any comment on whether or not it is a meaningful question to ask in the noncommutative case, but hopefully someone reading can comment on that.

As for the final question: Suppose $R$ is a local ring that is regular, with maximal ideal $M$. Then $M$ is prime, and by the definition of regular rings $R_M=R$ is a regular local ring.

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@MannyReyes Is there a type of regular local ring that makes sense for noncommutative geometry? – rschwieb Dec 12 '12 at 15:20

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