## Noncommutative Localization of a Ring : Complete Construction

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 StenstrĂ¶m (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)?

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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.

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If you don't have Ore conditions, how do you add? – Sasha May 21 2012 at 15:50
Thanks; I've made an edit. – Martin Brandenburg May 21 2012 at 20:09
Thanks for the link! But in chapter 6, Goodearl & Warfield only treat the case where the set S contains no zero-divisor. I am looking the more general case where S can contain zero-divisors as well. – SB May 21 2012 at 21:05

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

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 Thanks David! Here as well, Artin only construct the ring of fractions for a set S containing only regular elements. This construction does not work in the general case, when we allow elements in S to be zero-divisors. – SB May 21 2012 at 21:09 Oops, my apologies! I think I just have a "Here be dragons" sign over the zero divisors situation in my mind, which resulted in my being blind to that key detail of your post, as a sort of psychological defense mechanism =]. I'll leave this answer up, since it is a nice reference anyhow. – David Jordan May 21 2012 at 22:32

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 $R\rightarrow RS^{-1}$.

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

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