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 :

- for $a \in R$ and $s \in S$, there exist $b \in R$ and $t \in S$ such that $at = sb$,
- 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)?