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Extended the entry Cohn localization now starting with the ring viewpoint. Urs: I hope you will now agree that it is justified to call it a localization of a ring $R\to \Sigma^{-1} R$.
One should also point out (see Andrew Ranicki’s slides in the references at Cohn localization and his papers,specially the series with Amnon Neeman) that universal localization is much used in algebraic K-theory. algebraic L-theory and surgery theory.
One should
So please do! :-) (in the entry!)
Oops. Error. Invert all matrices not entries, of course.
If we have just a set $S$ of elements in $R$ and want to invert them, we take as $\Sigma_S$ the smallest set of matrices contining unit matrices with elements in $S$ such that if $A,B\in \Sigma_S$ and $C$ another matrix so that we can make a triangular block matrix with $A$ and $B$ as diagonal block matrix, then this triangular matrix is also there. Now inverting all such matrices is a little stronger requirement than to invert just elements in $S$. Thus $\Sigma_S$-inverting is stronger property than $S$-inverting, hence the category is smaller so there is a different chance for having an initial object, and indeed, it always exists, by the result of Cohn. Making the nondiagonal triangular block matrices amounts to be using Gauss elimination procedure to be solving systems of linear equations over a noncommutative ring with matrix operations from one side only. One side only is in a spirit of Ore, but it has the Gauss elimination procedure advantage over the scalar case.
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