Talk:Ring (mathematics)

message: 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)
thread_title:  "Ring action" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: 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)
replies: text: 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)
thread_title:  Undue weight? 

message: Prompted by : 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)
replies: text: I found a reference, which I added to the history section of ideal. Ebony Jackson (talk) 05:30, 3 January 2021 (UTC)
replies: text: 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)
thread_title:  Etymology of "ideal" 

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

message: 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)
replies: text: 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)
thread_title:  Erroneous example