Talk:Ring (mathematics)

"Ring action" listed at Redirects for discussion

A discussion is taking place to address the redirect Ring action. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 November 24#Ring action until a consensus is reached, and readers of this page are welcome to contribute to the discussion. D.Lazard (talk) 17:06, 24 November 2020 (UTC)

Undue weight?

Poonen's argument "it is natural to require rings to have a 1" suffers from being an unnatural argument: it unnecessarily posits an infinite axiom system just to provide a context in which to argue that it is "natural" to extend this construction backwards, without arguing for a gain in utility and without showing an understanding of the pitfalls of working with triviality. As far as Google Scholar can tell, this arXiv paper has only incidental citations by two other arXiv papers in the two years since its publication. I suggest that the weight given to Poonen's argument is completely WP:UNDUE in this context. —Quondum 23:05, 25 December 2020 (UTC)

I can see from the lack of response that this has effectively no support, so I will consider myself outvoted. —Quondum 01:58, 29 December 2020 (UTC)

Etymology of "ideal"

Prompted by this: Noether has a footnote (courtesy of Google translate): "Ideals are denoted with capital German letters. ${\mathfrak {M}}$ is intended to recall the example of the ideal of polynomials, commonly referred to as a 'module' or module of forms." This may relate to the etymology, since it is a footnote to the general definition of a (left) ideal denoted ${\mathfrak {M}}$. Would the context of polynomial rings give any hint about this? —Quondum 02:29, 3 January 2021 (UTC)

I found a reference, which I added to the history section of ideal. Ebony Jackson (talk) 05:30, 3 January 2021 (UTC)

In fact, before Hilbert, ideals that were considered were only those of number theory and Dedekind domains. This is still reflected by a part of terminology, such as the concept of a fractional ideal. The first use of ideals outside number theory seems to be Hilbert's basis theorem. I believe also that modules which are not ideals were introduced as "syzygy modules" by Hilbert in the same famous paper, for the proof of Hilbert's syzygy theorem. D.Lazard (talk) 09:35, 3 January 2021 (UTC)

Terminology

As I noted at Talk:Ideal (ring theory)#rng/ring confusion, there seems to be some fuzziness in the terminology in this area. Settling further conventions in WP might be helpful. For example, while ring (without qualification) has been settled as being unital and associative when used in WP, it seems to me to be pretty evident that the subject area Ring theory includes all theory of rngs as well as of the specialization to rings (but that article creates the impression that it is restricted to the latter). Given that "ring" has, and still is, used to mean either depending on the author or context, there must be innumerable examples of these. What I would like to guard against is creation of incorrect understanding due to the name being used in a way that it is interpreted differently from in the source.

A minor example: in this article, there is mention of what comes across as a dispute: whether a "ring" should or should not have a '1'. These are simply two classes of object, both valuable, and hence we need to be able to refer to each of them (fortunately settled in this narrow case). The changing use of the same term leads to issues with incautious editing if one attaches meaning to terms instead of the other way around.

I would suggest a careful review of many articles in ring theory with this in mind, starting with choosing one or more terms to start treating more consistently from a MoS perspective, maybe starting with "ring theory"? —Quondum 20:03, 3 January 2021 (UTC)

As a start, I added a note to Ring theory to mention rngs. Ebony Jackson (talk) 21:51, 3 January 2021 (UTC)

Thanks – that already is helpful. Also nice to know that I'm not totally out on a limb —Quondum 22:49, 3 January 2021 (UTC)

Erroneous example

Following up on the edit I just made removing the Atiyah and MacDonald text as an early example of a unital definition of rings. The text defines rings without a multiplicative identity, before adding that “We shall consider only rings which are commutative … and have an identity element.” It then states that it will gloss ‘commutative unitary ring’ as ‘ring’ for the rest of the text. I think that this is substantively different from what the other examples listed do, which is to just define multiplication as associative and possessing an identity.

This does produce another problem, which is that the claim that authors were defining rings as requiring multiplicative identities “as early as the 1960’s” now has no citation. I don’t know enough about the development of the topic to argue for or against that claim, but it seems to me this text represents an intermediary step in the development of the definition, where lip service is paid to Noether’s convention, but only unital rings are of interest. Lnkov1 (talk) 05:16, 13 June 2023 (UTC)

@Lnkov1: I agree about the general existence of unclearity. (Also see the last point in next section.) Bourbaki (which was pretty influential) does include unitarity in its definition (Algèbre I.§8.1), but I think this came a little later.JoergenB (talk) 22:15, 2 November 2023 (UTC)

Rather unclear formulation implicitly including unitarity

At present, there is a paragraph which implicitly seems to indicate that the only noteworthy alternative definition around demands unitarity but not associativity:

Although most modern authors use the term "ring" as defined here, there are a few who use the term to refer to more general structures in which there is no requirement for multiplication to be associative. For these authors, every algebra is a "ring".

Now, as noted earlier in the section, many authors employ "ring" as not necessarily being unitary ("having a 1"). Actually, so do also the EOM, MathWorld (https://mathworld.wolfram.com/Ring.html), and PlanetMath (https://planetmath.org/ring). I do not know if it a mistake or not to neglect those who do not demand unitarity in the quoted paragraph. If it is by intent, then there should be some better support earlier in that section, Ring (mathematics)#Definitions (or in the earlier section Ring (mathematics)#With or without unit) for the claim that "most" authors include unitarity in the ring definition itself.

Else, the paragraph should be modified, forinstance to

Although most modern authors use the term "ring" either as defined here or just without demanding the existence of a unit, there are a few who use the term to refer to more general structures in which there is no requirement for multiplication to be associative. For these authors, every algebra is a "ring".

Two remarks:

• I do not mean that we should change our use "ring" as "unitary ring" in the ring theory articles. This is a sufficiently common usage; and well suited for most of this theory. However, we must distinguish our choices of terminology "for practical purposes" from the description of the various general practices. (@Quondum: I note that unitary rings are "definable with a finite number of axioms" within the context of universal algebra, while e. g. fields are not. I'm not saying that the one or the other definition is "more natural", though. On the other hand, I'm a bit unhappy with the term "rng", since neither the r nor the nasal ŋ in general are employed as vowels in modern English, while our texts should be pronounceable; but surely there are alternatives.)
• Many times the author or authors of an article specify "local definitions" of some terms, for that article. This may be done rather briefly, by a sentence, or just a parenthesis, in the beginning of the article:

In this article, all rings are commutative, unitary, and noetherian. or

Let $A$ be any (unital but not necessarily commutative) ring.

This is part of the mathematical way of thinking, and quite all right. We have rather large possibilities to make local definitions; and we very often need them. I've seen some extremal examples of this (like an article where the addition in "a ring" was not necessarily additive); as long as the local definitions employed are clarified, this yields no real trouble (but in the case with the non-commutative addition, if I had been the referee, I might have suggested some changes of terminology). I suspect that the prevailing such localised use of "ring" in modern times does include unitarity; but this does not mean that the same authors necessarily employ this local definition "globally", in all other contexts. JoergenB (talk) 22:06, 2 November 2023 (UTC)

References

JoergenB, it is not clear to me why you choose to address that comment to me specifically. I have no in-principle issue with your statement about universal algebra.

On "rng", the pronunciation apparently includes an implicit vowel, thus being natural in English, albeit with an unusual spelling (but those abound, such as "cwm"). Do we have a better term to use here? I don't see one.

I agree that the statement creates a false dichotomy. I have tweaked it in a fairly natural way to avoid this. I am not convinced about the claim of the sentence though: "ring" can be used in its restrictive (unital associative) sense while the longer phrase "nonassociative ring" could mean a more general structure, i.e. with the axiom of associativity dropped. Your comment about 'local definitions' is valid, and EoM's use is hardly a reference for any given usage. I would be inclined to remove the statement altogether. —Quondum 02:32, 3 November 2023 (UTC)

@Quondum: Thanks for fixing the false dichotomy issue!

• My reason for addressing you about the finiteness of the axioms for a (unitary) ring was just that I felt a bit unsure of what you meant in your rebuttal of Poonen supra. (I agree with you that that article seem to be given undue weight in our article; I looked at its arXive version, and was not very impressed. However, their main point does not concern infinite products of rings, but rather is an argument for monoids being more natural than semigroups.)
• As for rng: I appreciate the way the word was constructed, in analogy with how the term versal was introduced as "universal except not demanding the uniqueness": Indeed, a "rng" is "a ring, except not necessarily containing a 'one'" (which may be written i in Roman numerals). However, while words as versal or morphism are reasonably easy to accomodate in the modern English language, I wouldn't state the same for rng.

It is true that many foreign words (including some names) borrowed into English usually retain their original spelling (as the Welsh cwm does). Now, since w in Welsh is regularly employed for a sound recognised as a vowel in both Welsh and English, IMHO, this is not a very good example. A more adequate one is offered by the city name Brno. As you can see, our article suggests an English pronunciation indeed involving a svarabhakti (or, in British English, even a complete replacement of the r), but also gives the Czech pronunciation (with a thrilled r indeed employed as a phonological vowel). There are various ways to insert and various qualities of svarabhaktis. One of the most well-established ones is the way the vocalic r in Sanskrit is represented in modern Indian usage, namely, by inserting a svarabhakti i after the r. Thus, the Sanskrit name spelt Ṛgwaida (with ai here originally standing for the diphthong in the modern English pronoun I) is pronounced Rigveda. If that svarabhakti construct were used for "rng", it would render the pronunciation ring (which hardly is intended, I hope).

In several cases (and even regularly in some verb paradigms), vowels in modern English derived from a svarabhakti being inserted. However, in the modern language, in most such cases, spelling at least gives some hint about the modern pronunciation of the resulting vowel sound. Introducing a new word like rng, without any such hint, is not a good idea (IMHO). I'd prefer going back to Bourbaki's pseudo-ring, if no-one has a better clearly pronounciable suggestion. JoergenB (talk) 18:26, 4 November 2023 (UTC)

Ah, my statement re Poonen from three years ago. I'm still tempted to remove the mention of Poonen altogether: saying that in the class of semigroups, loops are prettier does not cut it for declaring the class of semigroups to be incidental/inferior/to be dismissed, and the argument even feels circular, and has no notability.

"rng" is an awkward construction, introduced as it was through its spelling. However, I don't see what there is to be done about it (but we should be open to alternatives that are widely used). My example was not perfect, agreed.

As an aside, I would posit that it is a logical misstep in universal algebra to use the 0-ary mapping to posit the existence of an element. It is easy to reformulate the definition of a mapping in this context to show how very sensitive this existence implication is to the interpretation of "mapping". An implication in this reformulation is that, for example, that the class of groups is not a variety, just as the fields are not. The class of associative quasigroups is, and it is just the class of groups but for one extra member. —Quondum 12:36, 5 November 2023 (UTC)

Are you saying that Garrett Birkhoff did a logical misstep when he introduced universal algebra? It is not to Wikipedia to change the definition of a variety. The standard definition of a variety, as given in every reliable source, includes 0-ary operations. Moreover, with your suggested change, the theory of varieties would become very poor, as excluding several important structures (monoids, groups and rings). If fields do not form a variety, it is not because of the existence of 0-ary operations, it is because of the existence of operations that are not everywhere defined (division and multiplicative inverse). D.Lazard (talk) 14:42, 5 November 2023 (UTC)

D.Lazard, despite my preface "as an aside", you presumably misunderstand the spirit of my aside, and seem to be too eager to dismiss a nonstandard perspective while clearly failing to understand it. Do you think that I would even consider misrepresenting the mainstream definitions in WP? I hereby close this aside for discussion (at least by me), as a talk page is not intended for side discussions of this nature. —Quondum 15:46, 5 November 2023 (UTC)

I improved the Poonen reference a bit.

I think that Poonen's main argument is valid and worth to mention - but it indeed is an argument, not a proof. There are also valid arguments for the other side. As for his ring product counterargument, I have a feeling that a careful analysis should show that this more supports the definition witout unitarity. So, forinstance, a von Neumann regular ring R is a product of matrix algebras, where both R and its matrix ring factors are unitary, but the product is taken in the category of not necessarily unitary rings (rngs). Indeed, any (unitary) ring A containing an idempotent i different from both 0 and 1 may be exhibited as a sum or a product of two unitary subrings B and C whose unit elements are i and 1-i, respectively. Now, Poonen implicitly would argue that the projection of A onto either B or C indeed respects units. Now, this is true; but IMHO discarding $B\simeq B\times 0\subset B\times C\simeq A$ as a ring monomorphism is not very natural.

On the other hand, I note that Grothendieck's 'local definition' in 1960 also demands a categorical (or universal algebra) property, with mappings ("in general" including inclusions of subrings) respecting the unit element:

“ Tous les anneaux considérés dans ce Traité posséderont un élement unité; tous les modules sur un tel anneau seront supposés unitaires; les homomorphismes d'anneaux seront toujours supposés transformer l'élément unité en élément unité; sauf mention expresse du contraire, un sous-anneau d'un anneau A sera supposé contenir l'élément unité de A. Nous considérerons surtout des anneaux commutatifs, et lorsque nous parlerons d'anneau sans préciser, il sera sous-entendu qu'il s'agit d'un anneau commutatif. Si A est un anneau non nécessairement commutatif, par A-module nous entendrons toujours un module à gauche, sauf mention expresse du contraire. ” — A. Grothendieck, Éléments de géométrie algébrique, Chapitre 0 (Préliminaires), the entire paragraph (1.0.1); Institute des hautes études scientifiques, 1960, Publications mathématiques, N^o 4

(My literal translation on the spot; @D.Lazard, did I get this right?

"All rings considered in this Work will have a unit element; all modules over such a ring will be assumed [to be] unitary; the ring homomorphisms are always assumed supposed to map the unit element to a unit element; except when the converse is explicitly stated, a subring of a ring A will be supposed to contain the unit element of A. We mainly will consider commutative rings, and when we will speak about a ring without [adding something to the effect of further] precision, it will be tacitly understood that it concerns a commutative ring. If A is a not necessarily commutative ring, by an A-module we always will mean a left module, except [situations with] explicit mention of the converse.")

This is the first example of the new 'definitions' in the sixties mentioned by Poonen. Like for the earlier Atiya-McDonald example in our article, this local definition was adapted rather much to the specific demands of a specific work, without making any claim or hint that these local definitions ought to be adopted in general. Both works certainly were influential; and both may have influenced the decision of the Bourbakists to change their "official" definition of ring in the "new edition" of their Algèbre, published (in the relevant part) 1970. I think that mentions of these two 'local definitions' could be relevant in the historical part. I also think that we should restore a more general mention of the variants already in the lead; e. g., the now outcommented explanatory footnote

Some authors only require that a ring be a semigroup under multiplication; that is, do not require that there be a multiplicative identity (1). See the section Notes on the definition for more details.

Moreover, much of the discussion about unitarity should be placed close to the rest of the definition notes. Since the "not necessarily unitary rings" indeed still abund (as can be seen in EOM, PlanetMath, and MathWorld definitions; vide supra), we should not pretend that this just is a lingering but essentially historical definition. JoergenB (talk) 22:03, 6 November 2023 (UTC)

Hi JoergenB, your translation of Grothendieck looks very good to me, though maybe "sauf mention expresse du contraire" (twice) could be translated instead as "unless explicitly stated otherwise", to avoid confusion with the notion of "converse" in maths.

Do you know of later works by Grothendieck in which he uses the word "ring" to mean "ring without the requirement of a unit"? That would support the claim that he did not intend for it to be adopted in general. One could ask the same about Atiyah and Macdonald after they wrote their book.

A good place to discuss Wikipedia's convention that rings have a 1 might be the talk page for Wikipedia:Manual of Style/Mathematics#Algebra, since there are many Wikipedia articles that rely on this convention. Ebony Jackson (talk) 02:52, 9 November 2023 (UTC)

@Ebony Jackson: Thanks for your comments! I'm sorry if I didn't express myself clear enough. As I wrote above, I do not mean that we should change our use "ring" as "unitary ring" in the ring theory articles [emphasis added]. There are good reasons for us to fix a consistent terminology in our math articles; and the use of ring as including unitarity and assiciativity but not commutativity indeed is one of the most common extant usages. (On the other hand, I'm unhappy with the term rng; and I might later come with a suggestion of change. If I do, I'll indeed follow your advice, and discuss it, or at least announce its discussion, at Wikipedia:Manual of Style/Mathematics#Algebra.)

My (present) issues are precisely with the article ring, and only rather superficially with other algebra articles. The point is not our choice of meaning of "ring", but our report of the usage outside Wikipedia. What I claim is that the present article has some small deficiencies in describing the history, and some larger problems in describing the actual usages to-day. A reader of the present version should easily get the impression that defining rings as not necessarily unitary was an old habit, which nowadays is abandoned by most writers, while a few have not (yet) changed their outdated mode of using the term "ring". I claim that this is an incorrect impression. Since "ring" is used also in "the older sense" by a number of present authors, and this 'older' sense is the one primarily chosen by the three on-line mathematical encyclopediae EOM, MathWorld, and PlanetMath, we should back off a bit here (and possibly restore some older formulations). The reader should be clearly informed that the usage of "ring" as an associative but not necessarily unitary or commutative entity is one of the competing ones in the mathematical world of today; but that we in Wikipedia prefer the one where also unitarity is demanded, and that hence this is what the rest of our article ring is about. Regards, JoergenB (talk) 20:30, 11 November 2023 (UTC)

OK, sorry for misunderstanding you!

Some of the online encyclopediae you mention look to Wikipedia for their content, so I think it would be better for us to follow not them, but peer-reviewed books written by distinguished authors, say authors who are members of the honorary societies of their respective countries or have won other major prizes. My experience is that most of them currently include a 1 in their definition of ring, or at least say early on that their rings will all have a 1 (for the sake of readers who are used to the older convention). Ebony Jackson (talk) 18:31, 19 November 2023 (UTC)