What is the definition of 'regular local' and 'regular' for noncommutative rings? 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.)
 A: 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.
A: Let $R$ be a ring with centre $Z$. H. Bass (Algebraic $K$-Theory, p. 122) and C. W. Curtis & I. Reiner (Methods of Representation Theory Vol. 2, p. 22) give the definition:

$R$ is right regular if every finitely generated right $R$-module has a finite resolution by finitely generated projective right $R$-modules. 

Bass then gives the following results:

  
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*If $R$ is right regular, then $R$ is right noetherian.
  
*If $R$ is right regular, then $Z'\otimes_Z R$ is right regular for every localisation $Z'$ of $Z$.
  
*Assume that $R$ is finitely generated as a module over $Z$. Then $R$ is right regular if and only if $Z_{\mathfrak{m}}\otimes_Z R$ is right regular for every maximal ideal $\mathfrak{m}$ in $Z$. 
  

