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expanded (Fraenkel, Steinitz) and slightly changed
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user9072
user9072

Hilbert's definition for ring is (paraphrasing): given a collection of algebraic integers, a ring is everything that can be written as polynomial functions with integer coefficients of this given collection. (As an aside, personally, I now finally understood the idea behind the name integral domain/'Integritätsbereich'; a number field is also called 'Rationalitätsbereich', so rational domain there, being everything one gets with rational functions and the integral domain is what one gets with integral functions. Added: I saw had I started to read MO earlier I could have learned this usage due to Kronecker was mentioned by KConrad on the question linked to).

NowThen, let us jumpit seems the first axiomatisation of some decadesnotion of ring is due to Fraenkel (J. Reine Angew. Math., 1915). I stress some notion, since it does not completely match current practise in that each element is either a zero-divisor or invertible (and while non-commuativity is allowed it is only in a somewhat restricted sense in that the two products must only differ by an invertible element). The guiding example seems to be rings of integers modulo composites.

Regarding the name 'Ring' (that paper is also in German) he credits Hilbert but says there is some deviation of the meaning.

By constrast, Steinitz in his earlier axiomatization of fields (J. Reine Angew. Math., 1910) also discusses 'Integritätsbereiche' (integral domains) with exatly the axiomatization common today. (comm. ring, with unit, no zero-divisors).

Then to 'Moderne Algebra' (1930) by van der Waerden (based on lectures by Artin and Noether). [To be precise, I could not look at the original edition, but only some later edition, I hope this did not change over time.]

There one finds 'Ring' defined, (essentially) as is downdone now, as a basic notion; without any discussion of the naming. [To be precise, a ring there has not necessarily a multiplicative unit element and the existence of additive inverse and neutral element is expressed together via imposing solubility of $a+X=b$ for all $a,b$.]

In addition, one also finds 'Inegritätsbereich''Integritätsbereich' there however already with a different meaning than 'Ring'; namely as commutaitve ring without zero-divisors (yet not necessarily with unit element, so somewhat deviating from current usage and Steinitz).

I do not know what precisely happened in between, but I think one can make an argument that the structure is now called ring because it is called like that in 'Moderne Algebra', and one can note that also the naming of integral domain survived. (Except for slight deviation with unit element, but which until today is not quite uniform.)

And, it seems reasonable to assume that the naming of Artin, Noether, van der Waerden as for Franekel is directly inspired by Hilbert. After all, a ring has (just) the main properties mentioned by Hilbert for his 'rings', closed under addition, subtraction, and multiplication. What I do not know is whether there is any earlier axiomatization of ring in (or at least closer than Fraenkel's to) the current sense. Fraenkel's

To sum it up, this is all but a 'definite' answer, but I hope it contains some relevant information. In my opinion, it could be difficult, possibly even impossible, to ascertain what precisely motivated the choice of name and even more so to really pin down why one name survived and another not. (Itsay, Integritätsbereich did, Rationalitätsbereich did not). It could however be interesting to research literature and in particular lecture notes, if existant, of the beginning 20th century to see the development in more detail.) 

Still, ring seems like a good word as there are some potential intuitions (this circling back and the residue classes), also it is short and was I think quite different from preexisting names.

A rough search in Zentralblatt (but really rough) suggests that the usage of ring (in this form) gained traction only in the 1930s somewhat supporting the idea that 'Moderne Algebra' (and preceding lectures) was relevant.

Hilbert's definition for ring is (paraphrasing): given a collection of algebraic integers, a ring is everything that can be written as polynomial functions with integer coefficients of this given collection. (As an aside, personally, I now finally understood the idea behind the name integral domain/'Integritätsbereich'; a field is also called 'Rationalitätsbereich', so rational domain there, being everything one gets with rational functions and the integral domain is what one gets with integral functions).

Now, let us jump some decades to 'Moderne Algebra' (1930) by van der Waerden (based on lectures by Artin and Noether). [To be precise, I could not look at the original edition, but only some later edition, I hope this did not change over time.]

There one finds 'Ring' defined, (essentially) as is down now, as a basic notion; without any discussion of the naming. [To be precise, a ring there has not necessarily a multiplicative unit element and the existence of additive inverse and neutral element is expressed together via imposing solubility of $a+X=b$ for all $a,b$.]

In addition, one also finds 'Inegritätsbereich' there however already with a different meaning than 'Ring'; namely as ring without zero-divisors (yet not necessarily with unit element).

I do not know what precisely happened in between, but I think one can make an argument that the structure is now called ring because it is called like that in 'Moderne Algebra', and one can note that also the naming of integral domain survived. (Except for slight deviation with unit element, but which until today is not quite uniform.)

And, it seems reasonable to assume that the naming of Artin, Noether, van der Waerden is directly inspired by Hilbert. After all, a ring has (just) the main properties mentioned by Hilbert for his 'rings', closed under addition, subtraction, and multiplication.

To sum it up, this is all but a 'definite' answer, but I hope it contains some relevant information. In my opinion, it could be difficult possibly even impossible to ascertain what precisely motivated the choice of name and even more so to really pin down why one name survived and another not. (It could however be interesting to research literature and in particular lecture notes, if existant, of the beginning 20th century.) Still, ring seems like a good word as there are some potential intuitions (this circling back and the residue classes), also it is short and was I think quite different from preexisting names.

A rough search in Zentralblatt (but really rough) suggests that the usage of ring (in this form) gained traction only in the 1930s somewhat supporting the idea that 'Moderne Algebra' (and preceding lectures) was relevant.

Hilbert's definition for ring is (paraphrasing): given a collection of algebraic integers, a ring is everything that can be written as polynomial functions with integer coefficients of this given collection. (As an aside, personally, I now finally understood the idea behind the name integral domain/'Integritätsbereich'; a number field is also called 'Rationalitätsbereich', so rational domain there, being everything one gets with rational functions and the integral domain is what one gets with integral functions. Added: I saw had I started to read MO earlier I could have learned this usage due to Kronecker was mentioned by KConrad on the question linked to).

Then, it seems the first axiomatisation of some notion of ring is due to Fraenkel (J. Reine Angew. Math., 1915). I stress some notion, since it does not completely match current practise in that each element is either a zero-divisor or invertible (and while non-commuativity is allowed it is only in a somewhat restricted sense in that the two products must only differ by an invertible element). The guiding example seems to be rings of integers modulo composites.

Regarding the name 'Ring' (that paper is also in German) he credits Hilbert but says there is some deviation of the meaning.

By constrast, Steinitz in his earlier axiomatization of fields (J. Reine Angew. Math., 1910) also discusses 'Integritätsbereiche' (integral domains) with exatly the axiomatization common today. (comm. ring, with unit, no zero-divisors).

Then to 'Moderne Algebra' (1930) by van der Waerden (based on lectures by Artin and Noether). [To be precise, I could not look at the original edition, but only some later edition, I hope this did not change over time.]

There one finds 'Ring' defined, (essentially) as is done now, as a basic notion; without any discussion of the naming. [To be precise, a ring there has not necessarily a multiplicative unit element and the existence of additive inverse and neutral element is expressed together via imposing solubility of $a+X=b$ for all $a,b$.]

In addition, one also finds 'Integritätsbereich' there with a different meaning than 'Ring'; namely as commutaitve ring without zero-divisors (yet not necessarily with unit element, so somewhat deviating from current usage and Steinitz).

I think one can make an argument that the structure is now called ring because it is called like that in 'Moderne Algebra', and one can note that also the naming of integral domain survived. (Except for slight deviation with unit element, but which until today is not quite uniform.)

And, it seems reasonable to assume that the naming of Artin, Noether, van der Waerden as for Franekel is directly inspired by Hilbert. After all, a ring has (just) the main properties mentioned by Hilbert for his 'rings', closed under addition, subtraction, and multiplication. What I do not know is whether there is any earlier axiomatization of ring in (or at least closer than Fraenkel's to) the current sense. Fraenkel's

To sum it up, this is all but a 'definite' answer, but I hope it contains some relevant information. In my opinion, it could be difficult, possibly even impossible, to ascertain what precisely motivated the choice of name and even more so to really pin down why one name survived and another not (say, Integritätsbereich did, Rationalitätsbereich did not). It could however be interesting to research literature and in particular lecture notes, if existant, of the beginning 20th century to see the development in more detail. 

Still, ring seems like a good word as there are some potential intuitions (this circling back and the residue classes), also it is short and was I think quite different from preexisting names.

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user9072
user9072

First, a minor correction to the question, but it seems somewhat relevant: ring is not called 'Zahlring' in German, ring in German is 'Ring.'

As mentioned Hilbert introduced the word 'Zahlring' but also in the same sentence as synonyms 'Ring' and 'Integritätsbereich' (so ring and integral domain, resp.); see the embedded document on math.SE mentioned above; the mathematical meaning is that of a ring of algebraic integers in a number field (not necessarily the (full) one. In particular, already in that document he frequently just uses 'Ring', and defines for example 'Ringideale'. I just browsed the first part of the 'Zahlbericht' where this happens, there seems no motivation for the name to be found, he just gives in a footnote that Dedekind calls this 'Ordnung' [order].

The idea that the name is motivated by 'circling back' might or might not be true. But I could not find any trace of it there. In particular, there seems to be no result close by regarding the fact that the powers $\alpha^n$ somehow 'circle back' to linear combinations (the idea mentioned by KConrad); also no analogy to rings of residue classes is drawn. (Of course, it is proved somewhere that such a 'ring' has a finite $\mathbb{Z}$-module basis but the way this is presented does not suggest any particular 'circling back' idea.)

Hilbert's definition for ring is (paraphrasing): given a collection of algebraic integers, a ring is everything that can be written as polynomial functions with integer coefficients of this given collection. (As an aside, personally, I now finally understood the idea behind the name integral domain/'Integritätsbereich'; a field is also called 'Rationalitätsbereich', so rational domain there, being everything one gets with rational functions and the integral domain is what one gets with integral functions).

He then right away comments that a 'ring' is thus closed/invariant under addition, subtraction, and multiplication.

So, perhaps it is a ring just since one does not leave it even if one moves around, say like a boxing-ring. Or, I quite like the idea presented earlier of 'Ring' also being used to describe (figuratively) a collection of people with a certain relation among them, a property this word shares with 'Gruppe' [group] and also 'Körper' [field, but literally body], both seem to have been established by then already. (Which also is somehow a partial response to why a ring is a ring even though it is not more ring-like than a group or a field; the later two already had a different name.)

Now, let us jump some decades to 'Moderne Algebra' (1930) by van der Waerden (based on lectures by Artin and Noether). [To be precise, I could not look at the original edition, but only some later edition, I hope this did not change over time.]

There one finds 'Ring' defined, (essentially) as is down now, as a basic notion; without any discussion of the naming. [To be precise, a ring there has not necessarily a multiplicative unit element and the existence of additive inverse and neutral element is expressed together via imposing solubility of $a+X=b$ for all $a,b$.]

In addition, one also finds 'Inegritätsbereich' there however already with a different meaning than 'Ring'; namely as ring without zero-divisors (yet not necessarily with unit element).

I do not know what precisely happened in between, but I think one can make an argument that the structure is now called ring because it is called like that in 'Moderne Algebra', and one can note that also the naming of integral domain survived. (Except for slight deviation with unit element, but which until today is not quite uniform.)

And, it seems reasonable to assume that the naming of Artin, Noether, van der Waerden is directly inspired by Hilbert. After all, a ring has (just) the main properties mentioned by Hilbert for his 'rings', closed under addition, subtraction, and multiplication.

To sum it up, this is all but a 'definite' answer, but I hope it contains some relevant information. In my opinion, it could be difficult possibly even impossible to ascertain what precisely motivated the choice of name and even more so to really pin down why one name survived and another not. (It could however be interesting to research literature and in particular lecture notes, if existant, of the beginning 20th century.) Still, ring seems like a good word as there are some potential intuitions (this circling back and the residue classes), also it is short and was I think quite different from preexisting names.

A rough search in Zentralblatt (but really rough) suggests that the usage of ring (in this form) gained traction only in the 1930s somewhat supporting the idea that 'Moderne Algebra' (and preceding lectures) was relevant.