Why $S^{-1}R$ is called localization, despite $S^{-1}R$ is not always local ring?

Question

Let $R$ be unital commutative ring, and $S\subset R\setminus\{0\}$ be multilplicative set. Then $S^{-1}R=(R\times S)/\sim$ where $(a,s)\sim(b,t)\Longleftrightarrow \exists u\in S\text{ s.t. }u(ta-bs)=0$ is ring, and their element $\overline{(a,s)}$ is often noted as $\frac as$.

$S^{-1}R$ is called localization of $R$ by $S$. I was quite confused by this word, because the ring obtained by this may not actually be a local ring(and this fact would be easily verified by let $S=\{1\}$ and $R$ be non-local unital commutative ring, such as $\mathbb Z$.)

So, why $S^{-1}R$ is called localization? Of course, I know that $S=R\setminus\mathfrak p$ for some $\mathfrak p\in\text{Spec}(R)$, then $S^{-1}R=R_\mathfrak p$ is local. But it cannot satisfy me...

Answer 1

Neither localization nor local rings had those names when they were first created. The concepts were introduced by algebraists, but the names we use today were introduced later by algebraic geometers.

Localization as a purely algebraic concept, with the purely algebraic name "ring of quotients", was introduced by Grell in 1927 and achieved its modern general definition by Uzkov in 1948. The concept of a local ring was introduced by Krull in 1938, called by him Stellenringe (= point ring, cf. Nullstellensatz) and included the assumption that the ring is Noetherian. Krull's work on local rings, and on commutative algebra more generally, was purely algebraic. The term "local ring" was introduced by Zariski in 1943.

Some history is on p. 213 of Nagata's book Local Rings:

The notion of rings of quotients was first studied by Grell, who treated only the case where $S$ consists merely of non-zero- divisors. The case was clarified by Chevalley. Chevalley defined the ring of quotients with respect to a prime ideal, then Uzkov generalized completely.

The links I put on the names above go to the original papers. If you don't read Russian, you can see the MathSciNet review of Uzkov's paper is here.

Here is the notation and terminology in these papers:

1. Grell's construction assumes the multiplicatively closed set has no zero divisors (see p. 499). He wrote $P$ for the ring, $G$ for a multiplicative subset with no zero divisors in $P$, and the resulting ring as $\frac{P}{G}$, which he called Quotientenring, or literally "quotients ring". He observed at the start that he was generalizing the way an integral domain can be enlarged to a fraction field, which was due to Steinitz. Grell's paper on localizing (in 1927) has no geometric motivation, which isn't a surprise since algebraic geometry had not yet been built up using commutative algebra.

2. Chevalley's construction (in 1944) is in his 2nd link above and allows the multiplicative set $S$ to be arbitrary except that $0 \not\in S$, but he assumes the ring is Noetherian. His 1st link above (in 1943) is the paper where the term "Noetherian ring" first appeared, and in both papers his motivation is explicitly from applications to algebraic geometry. He wrote $\mathfrak o$ for the ring, $S$ for the multiplicative set, and $\mathfrak o_S$ for the result, calling it a "ring of quotients". That is a direct translation of Grell's terminology.

3. Uzkov's construction, which is developed purely algebraically, removes all restrictions on the multiplicative set and the ring except for the minor requirement that the multiplicative set doesn't contain $0$. Uzkov wrote $R$ for a completely general nonzero (commutative) ring, $S$ for a multiplicative subset of $R$, and $R_S$ for the result, which he called кольцо частных = ring of quotients = ring of fractions. As a reflection of how new Chevalley's term "Noetherian ring" was in 1943, Uzkov in 1948 doesn't use it, and instead speaks of Chevalley having worked with rings that satisfy the ascending chain condition on ideals.

There is an old MO question asking about the origin of the term "localization" and its systematic use as a technical tool here. After reading that, I suspect the term "localization" was introduced in the 1950s, inspired by its usage in algebraic geometry.

In summary, localization and local rings were each first created within algebra under different names (in 1927 and 1938, respectively). Although localization was created before local rings, the name "local ring" was introduced before the name "localization" and both were due to algebraic geometers (Zariski in 1943 and probably someone in Bourbaki in the 1950s).

Answer 2

You can flip this around and ask why is a ring with a unique maximal ideal "local"?

If you think of R as functions on some space X (which algebraic geometry tells you to do even if some rings might make that look, at first sight, like a questionable decision) then $S^{-1}R$ is the ring of functions on $X\backslash Z_S$ where $Z_S$ is the union of the zero sets of the functions in $S$.

To make this a bit more concrete, suppose that R is the ring of polynomials in $n$ variables over $\mathbb C$. Then each point in $\mathbb C^n$ corresponds to a maximal ideal $m_x$ (the kernel of evaluating a polynomial at that point).

Now you can localize at any multiplicatively closed set $S$, and if you take $S= R\backslash m_x$, then $S$ is multiplicatively closed and (because the points of $\mathbb C^n$ correspond to maximal ideals of R) localising at S throws away all the points of $\mathbb C^n$ apart from $x$, and the ring $S^{-1}R$ has a unique maximal ideal - the ideal generated by $m_x$, or equivalently, only one closed point. Thus a "local ring" is what you obtain by localising at a "point" of your space, but the operation of localisation is more general.

Answer 3

One cannot expect that mathematical terms involving "local" are directly related. The word is too general. We have "locally Euclidean spaces", "local rings", "localization of rings", etc. There is an "overload" of the word local, see for example this post.

The term "localization" for rings originates from algebraic geometry and is about studying varieties "locally" near a point $p$. The ring of function of this variety $R$ has a localization $S^{-1}R$, which is only a local ring if $S=R\setminus P$ for a prime ideal $P$ of $R$. So a "local" ring is not the same as a "localization" of a ring.