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A discussion is taking place to address the redirect
Ring action. The discussion will occur at
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D.Lazard (
talk)
17:06, 24 November 2020 (UTC)
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)
Prompted by this: Noether has a footnote (courtesy of Google translate): "Ideals are denoted with capital German letters. 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 . Would the context of polynomial rings give any hint about this? — Quondum 02:29, 3 January 2021 (UTC)
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)
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)
At present, there is a paragraph which implicitly seems to indicate that the only noteworthy alternative definition around demands unitarity but not associativity:
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
Two remarks:
References
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 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, No 4 |
(My literal translation on the spot; @ D.Lazard, did I get this right?
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
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)
|
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
|
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 10 sections are present. |
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)
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)
Prompted by this: Noether has a footnote (courtesy of Google translate): "Ideals are denoted with capital German letters. 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 . Would the context of polynomial rings give any hint about this? — Quondum 02:29, 3 January 2021 (UTC)
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)
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)
At present, there is a paragraph which implicitly seems to indicate that the only noteworthy alternative definition around demands unitarity but not associativity:
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
Two remarks:
References
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 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, No 4 |
(My literal translation on the spot; @ D.Lazard, did I get this right?
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
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)