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.)
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.