Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.)

share|cite|improve this question
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

1 Answer 1

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.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.