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Self-explanatory title really! A student today asked me why they were called integral domains -- and I realised that the word "integral" seems to be being used in a way totally unlike any other way I hear it used in mathematics. The student suggested that "integral" was used because the integers were an example -- but I didn't buy this because by that logic they could have been called "rational domains" or "real domains".

Any ideas anyone?

Conifold
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Kevin Buzzard
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    Here's a quote from Earliest known uses..., though it's almost surely not the earliest: `INTEGRAL DOMAIN is found in 1911 in Monographs on Topics of Modern Mathematics: Relevant to the Elementary Field by numerous writers: "Similarly, the integral domain [R1, R2, R3, ...] may be defined by replacing the expressions rational functions and rational coefficients in the preceding definition by integral functions and integral coefficients respectively."' – Bill Dubuque Jun 17 '11 at 18:35
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    I don't know if it is the origin, but in Dedekind's Sur la Theorie des Nombres Entiers Algebriques (1877), he writes "...it is easy to see that, in the domain of all integers we are considering at present, primes do not exist" (refering to the ring of all algebraic integers), and refers later to "the domain $R$" to refer to a system built up from the rationals; he uses "domains" throughout the work to refer to subrings of algebraic numbers (at least in Stillwell's translation). – Arturo Magidin Jun 17 '11 at 18:39
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    The term "integral" comes from rings of algebraic integers - the study of which motivated the abstraction of many algebraic structures (rings, domains, fields, modules, groups, etc), e.g. see Kleiner's historical exposition. – Bill Dubuque Jun 17 '11 at 18:39
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    @Bill: the term "integral over X" still now means "satisfies a monic poly with coefficients in X". I agree that this notion motivated all sorts of things. What I cannot see is how a historian can argue that "integral over the integers" (a very specific kind of complex number -- namely an algebraic integer -- these seem to be what the 1911 quote refers to) became "no zero divisors". That's the jump which I'm finding it very hard to make. – Kevin Buzzard Jun 17 '11 at 19:05
  • @Arturo: same comment to you. You are giving a good historical explanation of how we got the modern notion of an integral extension of rings -- but I don't see how the word integral became attached to "no zero divisors" -- because, for example, the subfields of the complexes generated by algebraic integers also have no zero divisors but are not integral in the sense being used in the pre-1930 reference at all. – Kevin Buzzard Jun 17 '11 at 19:07
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    @Kevin: I see; sorry for misunderstanding. I will point out that for "rational domains", the problem is that "domain of rationality" or "rational domain" was in fact an early name for "number field" (Hilbert mentions this in section 1 of the Zahlbericht, attributing it to Dedekind and/or Kronecker), named because it was constructed as "the collection of all rational functions of [a finite number of arbitrary algebraic numbers]". – Arturo Magidin Jun 17 '11 at 19:17
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    @Kevin "Integral" is overloaded in its historical evolution. It refers both to the notion of integral algebraic elements and also to subrings Z of fields Q which are considered to be "integers" (which yield an associated notion of divisibility: a|b in Z iff a/b in Z). But divisibility theory is much more complex in the presence of zero divisors (e.g. the notion of associate bifurcates into a few inequivalent concepts). So "integral domain" evolved to mean rings with amenable divisibility theory like the integers. – Bill Dubuque Jun 17 '11 at 19:21
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    But on the other hand "integral" in the sense of an extension of rings being integral and "integers" in the sense of "integers of a number field" are, in my mind, really closely related notions (the integers of a number field are just the integral closure of Z in the field, that is, the maximal integral extension). You're right about the notion of divisibility both (a) being dull in a field and (b) being broken in the presence of zero divisors -- and I think that this is a second possible answer -- independent of Qiaochu's. He suggests that the term might have come via "subring of a field"... – Kevin Buzzard Jun 17 '11 at 19:25
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    ...and you are now suggesting it's coming via the notion of "where is a good place to think about divisibility?". This also sounds like a reasonable hypothesis! As far as I can see, both your hypothesis and Qiaochu's remain unproved, and it might take some more digging to find out which is closer to the truth -- but I see both ideas as being plausible explanations. – Kevin Buzzard Jun 17 '11 at 19:26
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    @Kevin That's what I meant by "overloaded". I think the terminology was motivated by both of these viewpoints (and possibly more). – Bill Dubuque Jun 17 '11 at 19:31
  • @Kevin See also Leo Corry's paper The Origins of the Definition of Abstract Rings. – Bill Dubuque Jun 17 '11 at 22:49
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    @Kevin: See my comment responding to Goodwillie's comment to the question http://mathoverflow.net/questions/35286/origins-of-names-of-algebraic-structures where I give a reference to Cox's recent book on Galois theory. There I suggested looking at section 12.3A but now I think it's better to look at 12.3A and 12.3B. The student who asked you the question will find that reference accessible (unlike, probably, the old German articles by Kronecker and others). – KCd Jun 22 '11 at 19:52

3 Answers3

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Edit: I only see now that most if not all of the points were already made in the comments, but since I provide links and excerpts I leave the answer here. What I think is the answer comes right at the end of this lengthy post.

Edit 2: Added some formatting, minor corrections.

Disclaimer: This is not a definitive answer but I traced the German word Integritätsbereich through some famous older texts in German and I reproduce here what I found. It essentially confirms what Bill Dubuque said in several comments. Let me stress that I only looked in the most obvious places, so there may be earlier usages or better references than the ones I give here:

1. Early usage in the sense of algebraic integers

In Hilbert's Zahlbericht (1897) we find the following passage (unfortunately I can't access the English translation, so I hope German will do), which I'm reproducing from page 121 of volume 1 of his collected works (Göttinger Digitalisierungszentrum GDZ):

Zahlbericht

The footnote to Integritätsbereich reads: Nach Dedekind "eine Ordnung".

As Bill Dubuque pointed out in a comment, the term Rationalitäts-Bereich appears in Kronecker's Grundzüge einer arithmetischen Theorie der algebraischen Grössen (1882) and there is the following passage mentioning Integritäts-Bereich (reproduced from archive.org) on pages 14 and 15:

Kronecker Grundzüge 1Kronecker Grundzüge 2

As far as I can tell, Kronecker stuck to his intention of not using "Integritäts-Bereich" and I found it only mentioned once more at the beginning of § 22 on page 84.

For the sake of completeness: Hilbert also mentions Rationalitätsbereich and refers to Kroneckers work above as well as to Dedekind's Supplement XI to Dirichlet's Vorlesungen über Zahlentheorie, titled Über die Theorie der ganzen algebraischen Zahlen, available in volume 3 of Dedekind's collected works (GDZ) both in German and in French. As far as I can tell the word Integritätsbereich is not mentioned there (as Hilbert mentions the word Ordnung is used instead) but, incidentally, we find Zahlkörper, rationale ganze Zahlen, and, of course algebraische ganze Zahlen (and many more terms) used in precisely the same way as they are used today.

2. Modern usage

The modern abstract notion can be found in Emmy Noether's work (where else?), for example in Idealtheorie in Ringbereichen, Math. Annalen 83 (1921), 24–66 (the underlined text appears this way in Springer's online edition).

Noether Idealtheorie in RingbereichenNoether Idealtheorie in Ringbereichen 2

Of course, this doesn't explain why exactly this property of algebraic integers should be isolated, not something else but maybe that's something that only Emmy Noether herself could answer.

Added: A. Fraenkel, Über die Teiler der Null und die Zerlegung von Ringen, Journal für die reine und angewandte Mathematik (Crelle's Journal) 145 (1915), 139–176 contains the following passage:

Fraenkel Teiler der Null

The fact that an integral domain embeds into its field of fractions which is constructed in analogy with the construction of the rational numbers out of the integers is explicitly mentioned here. This seems a reasonable explanation for the choice of terminology which allows us to think of the elements of an integral domain as integers in some field. Of course, the term "ganze Zahl" (integer, literally: "entire number") is to be understood as "not a real fraction", that is: a fraction representable as $r/1$ with $r$ an element of the domain $R$.

In fact [slightly paraphrasing], Fraenkel even goes so far as to say that the ring properties of an integral domain are only artificial ("nur künstlich") and inherited from a surrounding field, as opposed to a ring with zero divisors which is deemed to posses a natural ("natürlich") structure of a ring. This also explains why "Integritätsbereich" is only mentioned this one time in the entire paper.

There is also a detailed discussion of the axioms of rings, quite close to what one finds in any basic algebra text nowadays.

t.b.
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    Thanks a lot. It seems to me that there is a case for arguing that Noether's use of the term should be a good candidate for "first time the term is used (with the meaning that it currently has)" on the history website. My reading of Hilbert and Kronecker is that, to the extent that they use the term at all, they use it to mean something different to what the phrase means today -- they are talking solely about algebraic integers as far as I can see. – Kevin Buzzard Jun 18 '11 at 15:58
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    @Kevin: Your reading of Hilbert and Kronecker is certainly correct. The way I read Kronecker is that he actually coined the term. I see your point about Noether but one should definitely examine (at least) Fraenkel's work on rings and Noether's earlier work more closely than I did. Fraenkel deserves being mentioned in my opinion because his paper marks at least a transition. – t.b. Jun 18 '11 at 16:24
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Earliest Known Uses of Some of the Words of Mathematics claims that the first recorded use of this term is in the 1911 text Monographs on topics of modern mathematics. It has a slightly different meaning there: if $r_1, ... r_n$ are numbers (I cannot get a clear sense of exactly what kind of numbers are considered here; perhaps algebraic integers) then $\mathbb{Z}[r_1, ... r_n]$ is called in that text the domain of integrity of the $r_i$ and $\mathbb{Q}(r_1, ... r_n)$ the domain of rationality of the $r_i$.

The terminology seems (based on my quick read) to have been motivated as follows: the first object is the closure of the $r_i$ under "integral operations" (addition, subtraction, multiplication, more generally composition with an integer polynomial) while the second is the closure of the $r_i$ under "rational operations" (addition, subtraction, multiplication, division, more generally composition with a rational function). Of course $\mathbb{Z}[r_1, ... r_n]$ is always an integral domain in the modern sense; perhaps that's how the terminology evolved.

van der Waerden's Modern Algebra (1930) contains the modern definition with no comment, so it was established by then (perhaps van der Waerden was responsible for establishing it!).

t.b.
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Qiaochu Yuan
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  • If I'd known about the history website I wouldn't have asked the question, probably -- but I am not at all convinced by the answer in some sense. If we have some algebraic integers then we get a subring of the complexes that they generate and this happens to be called an integral something -- but how do we get from this to the definition of an integral domain, where something rather different from "subring of the complexes finite over Z" is being stressed?? I guess what I'm saying is that "domain of integrity" seems to me to be a completely different notion than integral domain. – Kevin Buzzard Jun 17 '11 at 18:51
  • In particular, looking at the history website -- it almost seems a coincidence that the phrase "integral domain" is being used in this context -- I would like to argue that here "integral" really is being used in the standard sense of "satisfying a monic poly with Z coeffts" rather than "no zero divisors". What I'm saying is that this might be the first time the phrase is used in a maths book -- but it's not being used to mean what it means today, it's just a coincidence that the two words landed next to each other. – Kevin Buzzard Jun 17 '11 at 18:54
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    @Kevin: I wouldn't be so sure. The key point here is that "domain of integrity" is being used to denote a subring of a field, and at some point I wouldn't be surprised if someone realized that was a basic property that distinguished these rings from other rings. Of course I could be totally wrong; I really don't know what happened between 1911 and 1930 and can't think of any obvious sources to check. – Qiaochu Yuan Jun 17 '11 at 18:56
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    Hah! I do see your point Qiaochu. I would argue that there really are elements of "integral" here that mean "integral over Z" but the point is then perhaps some people used integral to mean what we now know as an integral extension, but others used it to mean "interesting subset of the complexes" that became "interesting subset of a field" that become "something with a field of fractions". Funny old world! I am still inclined to say however that the "integral domain" we see in the 1911 quote is a very long way from the notion as we now understand it. – Kevin Buzzard Jun 17 '11 at 19:03
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    If memory serves correct the terminology stems from Kronecker's famous Grundzuge - where he presented his theory of divisors. Here he called fields Rationalitatsbereich. This was often translated as "domains of rationality". For integral subrings he used the prefix ganze (whole or integral). Combining the first two terms yields "integral domain". – Bill Dubuque Jun 17 '11 at 19:08
  • There is a difference between the question "when did the words 'integral' and 'domain' first appear next to each other in the mathematical literature" (a question which we seem to have established the answer to) and "why does putting the words next to each other nowadays mean 'no zero divisors' rather than any of the other interesting properties that rings of algebraic integers have?". I think I would argue that the entry in the history website might be correct from some pedantic point of view in that "integral" and "domain" are next to each other, but I also claim that it does not get to the – Kevin Buzzard Jun 17 '11 at 19:14
  • heart of my question. As Qiaochu says, there might be a missing ingredient between 1911 and 1930. But for me the entry on the history website is a red herring. Qiaochu's comment ("subring of a field?") is interesting but is only, as he says, an extrapolation. Bill: you seem to be talking about the other notion of integrality -- that of satisfying a monic polynomial with coefficients over a certain base. This for me is barely, if at all, related, to the modern notion of an integral domain. – Kevin Buzzard Jun 17 '11 at 19:16
  • @Kevin: I think Bill is likely to be correct about the origin of the 1911 term. The exact connection between the 1911 term and the 1930 term, as you say, remains to be elucidated. – Qiaochu Yuan Jun 17 '11 at 19:19
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    Qiaochu: I am sure Bill is bang on the money re the 1911 term. I am still not at all convinced that the entry on the history page is at all useful, in the sense that the phrase happens to be being used, but not at all in the sense that it's used now. In the comments under the question Bill suggests another link between algebraic integers and no zero divisors -- the intermediate notion of "where is a good place to generate interesting questions about division?". As you already said, perhaps one needs to dig about between 1911 and 1930. – Kevin Buzzard Jun 17 '11 at 19:28
  • Remark for readers unfamiliar with Kronecker's divisor theory. Informally, he represents gcds as contents of (multivariate) polynomials, i.e. gcds of the coefficients of polynomials. So he works in polynomial rings over Z and their quotient fields ("rational functions"). Here he uses the prefix ganze (German for whole or integral) to distinguish (formal) polynomials from their fractions - which he calls rational. So polynomials are integral rational functions, and rings of polynomials are integral domains (of rationality), what he calls ganze Rationalitatsbereich. – Bill Dubuque Jun 17 '11 at 19:52
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I've always heard that the term Integral Domain comes from "domain of integrity" meaning "no cracks" in the ring, with the idea that zero divisors are like flaws in a diamond. This description comes from my teacher, the late great Prof. Gerhard Hochschild of UC Berkeley. So there.

Dan Bach
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