Talk:Ring theory

comment by Tarquin

Euclidean domain => Principal ideal domain => unique factorization domain => integral domain => Commutative ring.

Is the above a hierarchy of inclusion? If so, use the subset symbol. -- Tarquin

It's inclusion, but it doesn't make much sense to use subset symbol, because there aren't any standard "symbols" for "the class of all Euclidean domains", e.g., unless you want to make up several just for this article. Writing "Euclidean domain contained in PID contained in UFD, etc." is misleading, because it makes it sound like a ED is set-theoretically contained in a PID, contained in a UFD, not the same. Revolver 02:08, 11 Jun 2004 (UTC)

A ring is called commutative if its multiplication is commutative. The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers. Commutative rings are also important in algebraic geometry

Except that the ring article contradicts this and requires commutativity, which adds credence to my belief that this shouldn't be required in the definition. Revolver 02:08, 11 Jun 2004 (UTC)

No, multiplication in rings is not generally required to be commutative. I double-checked this in Herstein's Topics in Algebra as a sanity check. Isomorphic 02:19, 11 Jun 2004 (UTC)

I was wrong, the ring article doesn't require it, it requires unity. My fault. (Although I disagree with unity requirement, as well, that's another matter.) Actually, I just disagree with these universal wikipedia definitions (rather than article by article basis).

There are important examples of rings that do not have a identity. Regardless, and identity can always be formally adjoined by using the adjoint of the forgetful functor from the category of rings with unity to the category of rings.

There is inconsistency with the definition of "Ring" in the main article about rings. As far as I know a ring is assumed to have an identity unless stated otherwise and not the other way around. —Preceding unsigned comment added by 192.115.21.171 (talk) 11:28, 29 May 2009 (UTC)

Vote for new external link

Here is my site with ring theory example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

http://www.exampleproblems.com/wiki/index.php/Abstract_Algebra#Rings

Patent nonsense

Sorry for exaggerating in my edit comment -- the patent nonsense was only in Wikipedia for a little over a day before I reverted it. (I misread the date.) I find it embarrassing, though, that someone who trusts Wikipedia asked me for an explanation of it.—GraemeMcRae^talk 04:48, 21 January 2006 (UTC)

Could use an example

I don't know much about ring theory, other than that it keeps popping up on wikipedia everywhere... Could someone give some examples of rings and the 2 binary operators?

The integers Z, the rational numbers Q, the real numbers R the complex numbers C all under their ordinary addition and multiplication. Square matrix rings over any of these previous examples are also rings with matrix addition and multiplication. Try to chase some of the links on the ring theory page to arrive at more specific pages: they are likely to have other concrete examples. Rschwieb (talk) 15:25, 14 November 2011 (UTC)

Too Many Math Articles In Wikipedia Suck

The first two sentences are awful. Repeating the idea that ring theory is the study of rings. Very informative - not. Too much premature jargon. — Preceding unsigned comment added by 86.27.193.180 (talk) 23:54, 17 December 2011 (UTC)

sorting out ring theory and ring (mathematics)

Discussion started (and will proceed) at Talk:Ring_(mathematics)#Sorting_out_ring_theory_and_ring_.28mathematics.29 Rschwieb (talk) 15:47, 7 February 2013 (UTC)

need a definition

What is the definition (or what are the axioms) of a ring? Are they clearly stated in this article? I couldn't find them in a quick review. If they are there, please excuse me for asking. However, I think the first section after the intro / lede should say something like, "A ring is a set with two binary operators (typically called "addition" and "multiplication"), satisfying the following: ... ." I think the article might be easier to read and understand if this kind of definition appeared early in the article. Someone with a ring theory book handy should be able to provide this with a reference. Unfortunately, I don't have such handy. Thanks, DavidMCEddy (talk) 03:46, 2 March 2016 (UTC)

There's a separate article at ring (mathematics). However, there's been discussion on whether ring (mathematics) and ring theory should be merged, see e.g. Talk:Ring (mathematics)/Archive 4#merging ring theory into this article and Talk:Ring (mathematics)/Archive 3#New picture, new text. – Tobias Bergemann (talk) 10:48, 2 March 2016 (UTC)

Table of Ring/Group properties

I added a table near the top of this article, giving a summary of the principal ring-like and group-like structures, and the distinctions between them. I think this table is a very valuable summary which should appear somewhere prominent on Wikipedia.

I chose the Ring theory article, as a ring is the most "advanced" mathematical concept in this table. I think this is an appropriate place to draw the line, because the various other specialisations of rings are, well, more specialised.

Other possibilities include Algebraic structure - however, there, the table risks becoming bloated by a complete taxonomy of all algebraic structures. That may be useful in itself, but rings and groups are very important and basic concepts that occur again and again, and they have an associated set of terminology (which is rather confusing, particularly to nonmathematicians who are unused to memorising large amounts of terminology).

User:D.Lazard reverted my contribution, simply saying "This does not belongs to this article". I think this was contrary to Wikipedia:Revert_only_when_necessary. I am disappointed that apparently D. Lazard didn't even consider where else this table ought to go.

I am going to revert D. Lazard's revert. I would like to ask D.Lazard to please discuss the matter here instead of simply reverting my change again. Ijackson (talk) —Preceding undated comment added 18:27, 16 May 2017 (UTC)

It seems to me it might more usefully appear either later in a number of pages in this field or as a page of its own, prominently wikilinked from here and others. As a side note, it's also a bit afflicted with Caps Disease. Pinkbeast (talk) 19:03, 16 May 2017 (UTC)

Again, this content is not about ring theory, but about algebraic structures. Moreover, we have already a template {{algebraic structures}}, which contains in a more compact presentation the list of related structure. Therefore I'll replace this table by this template. D.Lazard (talk) 08:27, 17 May 2017 (UTC)

The template {{algebraic structures}} does not contain the same information. It is just a list of various structures and does not state the critical properties which distinguish them. Also, as I have explained, there is good reason to focus this information only on the structures culminating in fields. The template has too many structures and too little information about each.Ijackson (talk)

Where do you think it should go specifically? Should we make a template out of it? I think it could usefully appear in the article Group_theory too, at least, and maybe in some of the specific articles. It's a big bulky to make into an infobox and I don't see how to shrink it without losing its usefulness. As for "caps disease", feel free to fix.Ijackson (talk) —Preceding undated comment added 14:27, 17 May 2017 (UTC)

Please sign your posts with four tildes (~~~~) instead of three.

IMO, your table belongs to Algebraic structure. For including it there, I suggests to remove the heading "Examples" as well as the headings of its subsections, and to make sections from the bold texts such as "simple structures", "group-like structures, "ring-like structures", ... This would allows adding some text explaining that this is a classification by the number of operations and their arity, and also adding tables, like yours for clarifying the relations between structures. When this would be done, one could also modify the template {{algebraic structures}} for linking phrases such as "ring-like structures" to the corresponding section of Algebraic structure. This would make the information that you want to add easily accessible from each article about a specific structure. D.Lazard (talk) 16:23, 17 May 2017 (UTC)

I think D.Lazard's assessment is correct. The table is a bit of an off-topic intruder in the body of the article, especially as the first paragraph. This is already a good justification to the rollback. Secondarily (just my opinion, that is) it is not a very good table. It seems to suggest that the two notations of addition and multiplication have something to do with the definition of these structures, but that isn't true. It's also half-empty, for whatever reason. Frankly, nothing like this appears in any mathematics textbook. It makes more sense to have structures split apart by number of operations (and I believe that others have successfully contributed such charts.) I do recognize what it is, however. I've drawn many such pictures while learning my way around new disciplines. And I understand it's nice to share such things, but quite often they are not suitable for a wiki math article: they tend to be shaped/organized according to the note-taker's personal ideas. Rschwieb (talk) 16:47, 18 May 2017 (UTC)

Examples are needed

the word "example" appears no more than three times in this article, yet examples of rings are what makes ring theory important. 西米露喜欢橙子 (talk) 21:33, 4 November 2023 (UTC)

The right place for examples of rings is in Ring (mathematics), an article linked to in the first sentence of this article. This article contains many examples that are not always called "examples", especially when they are sufficiently important to be the subject of a specific section. D.Lazard (talk) 11:01, 5 November 2023 (UTC)