To recall: The edits around #24 were to clarify that the existence of the localization at any subset is a special case of the result of *Cohn localization* for which a fair number of references are given.

Madeleine,

One of the answers to this mathoverflow thread states that the localization of a ring always exists due to abstract nonsense. Quote,

]]>The localization $R S^{-1}$ always exists due to abstract nonsense: The subfunctor of $\mathrm{hom}(R,-)$ of homomorphisms mapping $S$ to units is continuous and the solution set condition is satisfied, so by Freyd’s criterion for representability the subfunctor is actually representable. Specifically, it consists of elements of the form $r_1 s_1^{-1} r_2 s_2^{-1} \ldots$, and sums of elements of this form.

Is there anything in the ring theory literature which explains the exact conditions under which a localization of an arbitrary ring away from an arbitrary multiplicative subset of the ring exists? Ore localization as defined via Ore sets covers cancellable multiplicative subsets in the general noncommutative case, but even in commutative rings it makes sense to talk about non-cancellable multiplicative subsets of the ring - if an element is a zero divisor and contained in the multiplicative subset in a commutative ring the localization at the multiplicative subset is a trivial ring.

]]>Jim Davis informs me that he has kindly looked into straightening out the statement in the entry (rev 28).

I went and have further streamlined the Idea-section and the References-section, trying to make clear that the references relevant for this entry here are those listed at *Cohn localization*

21: I looked into the history of the page, there are “anonymous” and some other past contributions to the page and I was far not the only writer. “Monomorphism” mistake is signed by the anonymous, if it was me (I do not remember) it was an obvious lapsus: in the subject we do spend much time dealing with zero divisors. In past I found some errors in published papers where some authors forget to check possible zero divisors (e.g. in the study of symbol map in certain algebraic context of filtered rings).

It is a matter of a taste – for the noncommutative rings – what generality to mean by localization of a ring and elsewhere by “noncommutative localization” (and rings of quotients as a third case where also some very different constructions like Martindale rings of quotients apply). For many, Cohn’s localization is nowdays equated to “noncommutative localization”, for some Gabriel localization of a ring is the localization of a ring, and for some still only Ore qualifies. Besides, the noncommutative localization may mean the localization of the category of modules or only the category of finitely generated projectives. While I was writing I had some tradition in mind, but after all these years, I can say that there are so many different communities with their conventions. In any case, for a localization of a ring one should start with a general ring and obtain a ring (as opposed to localizations of categories of modules) together with a canonical morphism or rings (component of the unit of adjunction for categories of modules, applied to ring itself). At the level of categories of modules we have an affine functor (adjoint triple) if additionally the forgetful functor from the localized category to the category of modules over localized rings is an equivalence. Many references pretend that Ore case has better properties than general biflat affine localizations (biflat means that both the localization and its right adjoint are flat, that is the localization considered as an endofunctor is also flat), but for most purposes this is not the case, and especially I see no difference in practice between one sided and two-sided Ore sets as far as (useful) properties of localization are concerned.

]]>In order to clarify this situation, I have deleted from this entry everything that pertains to commutative rings and merged that material into the entry *localization of commutative rings*.

To the Definition that remains here, I have added the warning that it’s existence is not guaranteed and pointed to *Ore localization* and *noncommutative localization*.

It seems that much of this scattered material on the noncommutative case could usefully be merged into one coherent entry.

]]>I don’t know about localization of non-commutative rings; I believe the only one who wrote about this here is Zoran. He lists more references at *noncommutative localization*, though I haven’t looked at these and don’t know how useful they are.

I see that part of the problem with the entry entry here is that I was editing cross-purpose with Zoran, being oblivious of his intended non-commutative angle; for instance when I added the definition of localization for commutative rings in rev 7. I only now see that pjhuxford #18 was looking into adjusting this.

]]>j and l need not be injective, for example if 0 \in S, then S^{-1}R is the zero ring.

Side comment: It would be nice to have a reference for the existence of the localization of a general noncommutative ring.

Jim Davis

]]>According to this Wikipedia article, the localization of a noncommutative ring cannot always be formed as described. The relation used to define the equivalence classes is not necessarily transitive in the non-commutative case. The provided reference to the Stacks project is in a section contained in the chapter on commutative algebra. That is why I made this change.

The Wikipedia article I linked mentions the Ore condition as a sufficient condition for this, which I see was mentioned earlier in this discussion, and has its own nlab page here: Ore localization. Perhaps a more nuanced discussion about when the given definition does apply to non-commutative rings is warranted.

]]>I removed the heading on noncommutative rings, and moved the corresponding definition of localization under the heading of commutative rings.

]]>It is universally applicable (for all subsets of $R$, unlike any other noncommutative localization); in that sense it is universal among all noncommutative localizations. (true, Gabriel’s localization is also for every $S$, via filter $\mathcal{F}_S$ but it does not really invert in such formal sense). Funny enough Cohn called it **“inversive localization”** in what I think is the first article he wrote on the topic; this is even worse terminology, comparing to most modern localization concepts which are, of course, also inversive.

- P. M. Cohn,
*Inversive localization in noetherian rings*, Communications on Pure and Applied Mathematics**26**:5-6, pp. 679–691, 1973 doi

(Unfortunately I do not have access to this article (neither from IRB nor from IHES). Anybody can send me the pdf ?)

However, there are also partial version where one inverts just matrices from the left (or right), which he called semi-invertive in this paper. They are of some use in algebra.

]]>I won’t further argue. But I record that I am opposed to using “universal X” in an entry title for something that is far from being universal among $X$s.

]]>Though the term is not the happiest, we do not intend to ever use the technical term universal localization for some other localization than Cohn’s, do we ? I mean many localizations have universal properties but we do not say that the localization itself is universal, so we do not need to free the term anyway… Besides, Cohn may have been motivated by the fact that he can localize for every $S$ and even $\Sigma$; other kind of taking quotients in ring theory needed special assumptions on inverting set, or did not have a universal property. This one is universal in three different senses – first in the sense for every $S$ and $\Sigma$ it exists, and besides it has a clean universal property at the level of rings; and it also has another universal property at the level of full categories of modules (because it is a torsion theory).

]]>Another interesting thing is that Cohn localization may be viewed (by a result of Vogel) as $H^0$ of certain Bousfield localization of a triangulated category.

]]>Read above. It is sometimes viewed as kind of a torsion theory and ring is not that clearly seen in that case. Sometimes one looks at localization of a ring and sometimes at associated functor. in both cases people call it universal localization or Cohn localization or Cohn universal localization. And it has some generalizations.

]]>So you are definitely opposed to renaming the entry to “universal localization of a ring”?

]]>In ring theory.

And in torsion theory of abelian categories, in skewfields, in universal algebra community, and in the setup of triangulated categories and in topology, and also in descent theory. See volume edited by Ranicki to get some feeling of how spread it is.

I agree that it is rather pompous to call it universal (it has clean universal property what some ring theoretic localizations do not, as I emphasise today!) as there are many other “universal” localizations in different setup. But it is named so by Cohn, and then accepted by algebraic topologists like Vogel, Ranicki, Neeman, by algebraists like Bokut and then it spread widely. I like to call it Cohn localization what is also somewhat used (in algebra but not so much in topology where universal is predominant for that notion).

]]>Universal localization is an entirely standard term.

In ring theory. You see, there are subfields of mathematics where even a bare “abc” is an entirely standard term, as we have just seen. But when we are talking to a potential audience that consists of more than a small group of specialists, we should try to use more self-contained terms.

Given all the entries that we have on the notion of “localization”, as you have listed above, given all the many many fields that these apply to, and given that about every notion of localization is universal in *some* sense, it seems unreasonable to declare that the words “universal localization” should refer exclusively to a very particular case of a localization in a very particular context only.

How about “universal localization of a ring”, at least?

]]>Urs, Cohn localization is yet a stub. Cohn localization is most often in ring theory viewed and used only as a localization of ring and not as its associated torsion theory/localization functor !

Namely, take a multiplicative set of matrices $\Sigma$ (let us not dwell on the definition of it, it is a set of matrices of varying sizes satisfying some closure properties) over a ring $R$.

We say that a map of rings $f: R\to S$ is $\Sigma$-inverting if all entries of $f(A)$ for each $A\in \Sigma$ are invertible in $S$. The Cohn localization of a ring $R$, is a homomorphism of rings $R\to \Sigma^{-1} R$ which is initial in the category of all $\Sigma$-inverting maps (which is the subcategory of coslice category $R/Ring$).

Universal localization is an entirely standard term. Notice that every subset $S\subset R$ can be viewed as generating some multiplicative set of matrices of various sizes (the need for latter is to fake the Ore property by sort of Gauss elimination procedure). The Ore localization is a special case or Cohn localization.

]]>Okay. Though calling Cohn localization a localization of the ring instead of of its category of modules is a bit of an abuse of language. But all right.

I have edited *Cohn localization* a little. Is it wise to have the entry titled “universal localization”? That seems a very ambiguous title for a very specific entry.

Thanks, though, again, as I explained earlier, this (what you added) is true for commutative localization and Ore localization but it is *not* appropriate for Cohn localization (where matrices are inverted) nor for Gabriel localization of a ring itself at a filter $\mathcal{F}_S$ obtained from a multiplicative set $S$ (not all kinds of noncommutative ring localizations, are having this ring theoretic universal property). I added a sentence modifying a bit this statement. I think more precise statement should not be in that entry, but rather at entries dedicated to special kinds like Cohn localization, Ore localization, Martindale localization, Gabriel localization…

I have formatted slightly at *localization of a ring* and added the half-sentence on what characterizes the localization map.

Entries with word localization in the title so far: Beilinson-Bernstein localization, Bousfield localization, Bousfield localization of model categories, Bousfield localization of triangulated categories, Gabriel localization, Ore localization, affine localization, [[algebraic microlocalization, cohomology localization, commutative localization, compatible localization,equivariant localization, equivariant localization and elimination of nodes, homology localization, homotopy localization, iterated localization, localization, localization of a commutative ring, localization of a model category, localization of a module, localization of a ring, localization of a simplicial model category, localization of an (infinity,1)-category, localization of an abelian category, localization of an enriched category, microlocalization, noncommutative localization, reflective localization, simplicial localization, simplicial localization of a homotopical category, strict localization, topological localization, universal localization

]]>I have created a new stub localization of a ring different from localization of a commutative ring. Note also field of fractions (which refers to commutative localization).

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