# Where does the letter S in "$S$-units" and in localization $S^{-1} R$ come from?

In number theory, we may encounter the notion of $S$-unit, $S$-integer, etc. where $S$ is a finite set of prime numbers (for simplicity). For instance, if $S = \{2,3\}$ then the $S$-integers are the elements of $\Bbb Z[1/6]$. I was wondering if there is a particular reason as to why the letter $S$ was chosen to systematically denote a finite set of primes. I wouldn't be surprised if it came from a German word (as many number-theoretic words).

My first thought that the $S$-integers are related to a localization of $\Bbb Z$ — as the example above shows. Usually, when $R$ is a commutative ring and $S \subset R$ is multiplicatively closed with $1 \in S$ (e.g. $S$ is the set of powers of $6$), we can construct a new ring denoted by $S^{-1}R$. My second question is: why did we choose $S$ as a letter? Is it just for "subset", or because $S$ is the next letter after $R$?

I don't know where/when/who first introduced these two notions, the one of $S$-integer, and the one of localization. I'm not sure that these two are historically related, at least in the choice of the letter $S$. I would be grateful to any piece of information about this.

Thank you very much!

• First, you should include $\infty$ in $S$ for your example: in number fields, $S$-integers are integers at all places outside $S$. Anyway, I think the concept of $S$-units preceded $S$-integers and first arose in the context of the $S$-unit theorem of Hasse and Chevalley. I hoped that searching for "S-unit" on MathSciNet would help, but it doesn't because it returns results like "Dirichlet's unit theorem" which literally has the term "s unit" as part of it.
– KCd
Jun 29 '18 at 21:59
• Not everyone uses $S$, e.g. Weil’s Basic number theory (1967) systematically has $P$. It seems not impossible that $S$ and $S$-unit first appeared in Artin-Whaples (1945, p. 487). Jun 29 '18 at 23:20
• Jul 1 '18 at 14:25

As Francois Ziegler suggests in his comment, the notation $S$ and term $S$-unit might go back to Artin and Whaples in their paper about the product formula: "Axiomatic Characterization of Fields by the Product Formula for Valuations" (Bull. AMS 51 (1945), 469-492). Here they write $S$ for a finite nonempty set of primes that includes all the archimedean primes and they define $S$-units on p. 487.

The $S$-unit theorem is in a weak form as Theorem 5 (p. 489) and in its full form as Theorem 6 (p. 491), with a footnote attributing the statement of the theorem to Hasse and the proof to Chevalley in Chevalley's paper on class field theory in Annals of Math. 41 (1940), 394-418. There Chevalley writes $E$, not $S$, for a finite set of "prime divisors containing all the infinite prime divisors" and the $S$-unit theorem is stated two paragraphs after Theorem 3. Since Chevalley's paper is the original appearance of the $S$-unit theorem, using notation no longer around today, it looks like the notation $S$ and the term $S$-unit are due to Artin and Whaples.

Even though the $S$-units are the unit group of the ring of $S$-integers, the concept of an $S$-integer came later since Artin and Whaples don't mention any such type of generalized integers in their paper. It's sort of historically fitting that $S$-units were created before the ring of which they are units, since Chevalley did the same thing for ideles: he defined that group (1936) and worked with it before anyone had defined the ring of adeles of which the ideles are the unit group. Strange but true. The Wikipedia page for adelic algebraic groups points out that Chevellay started using the term "idele" in 1940, while the adeles were called repartitions and (Tate's thesis) valuation vectors before the term "adele" became standard later in the 1950s.

• Thank you very much for your wonderful answer. Do you think I should ask a new question for the localization, since it has apparently nothing to do with the $S$-units, $S$-integers, ...? Jun 30 '18 at 8:59
• (By the way, do you know where the word "adele" first appeared? Apparently, idele comes from "ideales Element", introduced then by Chevalley as you mention. I've read on Wikipedia that "adele" means "additive idele", and is related to Bourbaki's philosophy…). Jun 30 '18 at 9:03
• Doing a search on MathSciNet (where reviews go back to 1940), the earliest appearance of adele outside of being a woman's name is Iwasawa's review MR0105410 in 1957, where he writes "so-called valuation vectors (adèles)", and Cartier's review MR0094362 also in 1957 uses no special term at all and he says a mouthful to describe this ring (see 1st sentence). In 1953 Iwasawa was saying "valuation vector" (see Tate's review of MR0053970). The Wikipedia page for adelic alg. groups has a link to numdam.org/article/SB_1954-1956__3__23_0.pdf from Dec. 1954 which says adeles. So 1954?
– KCd
Jun 30 '18 at 10:54
• Cartier was a member of Bourbaki 1955-1983, so if he wasn't using "adele" in that 1957 review then it really wasn't even yet standard in French.
– KCd
Jun 30 '18 at 10:57
• Concerning localization, the Historical Note section at the end of Bourbaki's Commutative Algebra says the idea originally goes back to Grell (Math. Ann. 97 (1926), 490-523, p. 510 according to the 1st footnote of Chevalley, "On the notion of the ring of quotients of a prime ideal," Bull. AMS 50 (1944), 93-97) and the general case is due to Uzkov in 1948. There is no "S" in Grell's paper for a mult. set but there is in Chevalley, who writes $\mathfrak o_S$. Uzkov uses $S$ and $R_S$, which is similar to Chevalley. It looks like $R_S$ preceded $S^{-1}R$.
– KCd
Jun 30 '18 at 11:37
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