Question

I've been looking for the following construction in the literature, but I've only been able to find (very) partial proofs or proofs of special cases.

Let $R$ be a non-commutative ring and $S$ a multiplicative subset (i.e., $1 \in S$ and if $a, b \in S$ then $ab \in S$, the set $S$ can in particular contain zero-divisors).

It is known that the localization $RS^{-1}$ exists if :

1. for $a \in R$ and $s \in S$, there exist $b \in R$ and $t \in S$ such that $at = sb$,
2. if $sa = 0$ for $s \in S$ and $a \in R$, then there exists $t \in S$ such that $at = 0$.

Many sources give the complete construction in the simpler case where the set $S$ only contains regular elements (i.e. non-zero divisors).

The general case is presented in (amongst others) : Rings of Quotients : An Introduction to Methods of Ring Theory by Bo Stenström (Prop. 1.4, Chap. II, p.51) or in Algebra, Volume 3 by P. M. Cohn (Thm. 1.3, Chap. 9, p. 350) but in both cases large parts of the proof are omitted.

Does anyone know where I can find the complete construction? In particular, the fact that the multiplication is well-defined (i.e., does not depend on the representing objects of the classes)?

Answer 1

The localization $RS^{-1}$ always exists due to abstract nonsense: The subfunctor of $\hom(R,-)$ of homomorphisms mapping $S$ to units is representable since it is continuous and the solution set condition is satisfied, so that we can use Freyd's criterion for representability. Specifically, it consists of elements of the form $r_1 s_1^{-1} r_2 s_2^{-1} \dotsc$, and sums of elements of these form. However, for practical uses, one wants elements of the form $r s^{-1}$ (or $s^{-1} r$, or both options) and an easy condition for equality of such fractions (some people put this into the definition of the localization, but this is artificial). This is contained in the Ore condition. You can find this everywhere (just google for "Ore condition"), for example in "An Introduction to Noncommutative Noetherian Rings" by K. R. Goodearl, Robert B. Warfield, Chapter 6.

Answer 2

If I remember well, the second chapter of

J. C. McConnell, J. C. Robson. Noncommutative Noetherian rings, vol. 30 of Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2001)

contains a rather detailled proof of the Ore's theorem.

Edit: I just checked it on Google books and they allow zero divisors as well. The point is that if $S$ satisfies the Ore's condition (which is nothing but your first condition), then the set {$r\in R, rs=0$ for some $s\in S$} is an ideal in $R$ which is precisely the kernel of the natural map $R\rightarrow RS^{-1}$.

Answer 3

Hi,

The following set of notes seems to do what you want. They are written by M. Artin (and posted on the website of P. Etingof, for a course he taught):

http://math.mit.edu/~etingof/artinnotes.pdf

Answer 4

Categories and Modules with K-Theory in View (Cambridge Studies in Advanced Mathematics) A. J. Berrick , M. E. Keating