# Talk:Ring (mathematics)/Archive 2

## Draft Version

There is another version of this article: Talk:Ring (mathematics)/Draft. (23 December 2008)

## Improvements

I am a bit worried about this article. Namely, it does not contain so many of the important concepts in ring theory. For example, a few hours ago, it contained nothing about morphisms of rings! I have made some improvements to the article and hope to get it to a higher standard; perhaps something like group (mathematics)? —Preceding unsigned comment added by Point-set topologist (talkcontribs) 17:30, 17 December 2008 (UTC)

You are absolutely right. Many of the (even top-importance) algebra articles contain little more than the mere definition. So, simply go ahead! Beware that there is also commutative ring (which is in slightly better shape, after some recent edits of mine), so it is important to have a clear picture of what belongs where. Jakob.scholbach (talk) 10:31, 18 December 2008 (UTC)
Looking at it more closely, I don't think that I can get it to Group (mathematics) so easily... But I will still try to get it to a decent standard. In particular, the article is lacking images.

Point-set topologist (talk) 10:40, 18 December 2008 (UTC)

I have added an image of a commutative diagram to the section on the definition. Could someone please position the image so that it is a bit lower (and is on the same row as the axiom for associativity of multiplication). Point-set topologist (talk) 10:51, 18 December 2008 (UTC)

You can move an image by moving it to the line (in the source code) where you want it to be. However, remember that a commutative diagram and tensor products will not be helpful to readers not acquainted to this (and adds, IMO, nothing to the understanding conveyed by the equations). Also, giving two times the definition is superfluous. Jakob.scholbach (talk) 11:02, 18 December 2008 (UTC)
Adding images is not the most important point at this stage, I think. What I can recommend as a working procedure: figure out what you want to be in the article, put the links in the "see also" section and start writing on the more important items in the see also section. Also, at an article like this, it is critical to stay very focussed, otherwise you end up deleting lot of content you wrote earlier. For example, the subring section will be too long when the article reaches maturity. Finally, it is also necessary to remember that Wikipedia is not a textbook. So, for example, instead of
A morphism of rings is simply a natural correspondence between two rings R1 and R2 in such a way that ring operations are preserved. The formal definition is given below.
Formal definition
I would write
A ring homomorphism is a map between two rings R1 and R2 in such a way that ring operations are preserved. Formally, ...
Hope that's not discouraging..., cheers Jakob.scholbach (talk) 11:11, 18 December 2008 (UTC)

Thanks Jakob; I fixed that up. I will do the same for the image when I get more time. I am just not really sure which concepts should be included in the article. For example, Group (mathematics) does not seem to have anything on the center of a group and from what I read, it looks like a top-standard article. So should this article contain anything about the center? Anyway, I think that ideals, quotients, subrings, morphisms and direct products are most important (and basic) in ring theory so I will try to expand the corresponding sections.

One more question: I am thinking of adding an example of a finite ring. The most obvious one would be of course the the integers mod n with its natural operations but I am not sure how to get an image of a 'ring table' for this. I noticed that there was one on Group (mathematics) for the dihedral group. Would it be possible to get an image like that?

Thankyou very much for your help.

Point-set topologist (talk) 14:53, 18 December 2008 (UTC)

I was just wondering: I am not really familiar with the policies but what class would the article be right now?

Thanks —Preceding unsigned comment added by Point-set topologist (talkcontribs) 14:56, 18 December 2008 (UTC)

How does the article look now? —Preceding unsigned comment added by Point-set topologist (talkcontribs) 20:13, 18 December 2008 (UTC)

## Informal review

As per p.s.t.'s request, I give some points which I think would improve the article further (currently, I think, the article is Start class).

• Be careful that you don't simply copy a textbook.

No, everything that I am adding is from the top of my head (which I am now backing up by references). Point-set topologist (talk) 20:48, 19 December 2008 (UTC)

• Focus on interesting, encyclopedic information. For example the proof of 0a = a0 = 0 is certainly not encyclopedic. If you absolutely want the reader to know about this, provide a precise reference to a book, including page number or chapter. This way, the information will be available, but does not clutter up the article.
Do you by chance know any good books on ring theory? I don't have access to a library in the near future so it will be difficult for me to do that. If you know any good books for citations (ref section is perfect), I would greatly appreciate it if you added those. Point-set topologist (talk) 20:52, 19 December 2008 (UTC)
• Repeating "Let (R, +, ⋅) be a ring. " every time is odd. Perhaps mention it once at the top (compare with commutative ring).
Maybe I will write: In this article, the notation (R, +, ⋅) will not be used for a ring since this is understood (rather 'let R be a ring'). But this is low priority at this point in time.
• Markup-style:
• bullets
• certainly
• don't
• make the text easier to read. Prefer true prose where possible (this is almost always possible).
• Cleanup tags don't help too much.
They are just a note to myself telling me what parts I should improve. Point-set topologist (talk) 20:52, 19 December 2008 (UTC)
• Motivate and explain more what is going on. (The motivation section is well-done in this respect). For example: what is the purpose of the quotient ring? (This is actually more important (IMO) than giving every little detail of all the definitions. This article is to give an overview of related notions. Hence it may have to be a bit short at times, if necessary even sloppy (but make clear what is sloppy if you need to be sloppy)).
This I definitely need to do (like one interesting (but simple) problem is to determine under what conditions a quotient ring of a ring is a field; this could be a motivation). Point-set topologist (talk) 20:52, 19 December 2008 (UTC)
• Provide more orientation to the reader, i.e. what are the main classes or types of rings (e.g. comm. / non-comm)? The examples list is pretty long, try to give it more structure by explaining the relations between the examples. When explaining the basic notions, come back to the examples.
• What should be included in the article? See [1] for WP articles linking to this one. This might give you some ideas on what to include. Other than that, looking into books, scientific articles, talking to people will give more. [2] contains a few other ideas. Also see [3] and the subpages listed there.
• Z/4 would be better represented giving multiplication and addition tables (compare with field (mathematics)).
• The article needs references. For the moment, if every bigger section has one, this would be good. I personally recommend using {{Citation}} templates or similar ones. [4] contains reference information for many math books.
I think that the reference section is now perfect. Point-set topologist (talk) 20:48, 19 December 2008 (UTC)

Jakob.scholbach (talk) 21:34, 18 December 2008 (UTC)

Thanks! I will respond to each of them once sorted. Point-set topologist (talk) 20:48, 19 December 2008 (UTC)

## Copied text that may be added later

General definition

More generally, for any index set J and collection of rings (Rj)jJ, there is a direct product ring. The direct product is the collection of "infinite-tuples" (rj)jJ with component-wise addition and multiplication. More formally, let U be the union of all of the rings Rj. Then the direct product of the Rj over all jJ is the set of all maps r : JU with the property that rjRj. Addition and multiplication of these functions is via the addition and multiplication in each individual Rj. Thus

(r + s)j = rj + sj and (rs)j = rjsj.
Wouldn't it be simpler to just say it's the cartesian product of the rings in question equipped with the pointwise operations? I think for this section we can assume the reader either knows what the cartesian product is or is willing to click on the link. Algebraist 20:28, 19 December 2008 (UTC)
I didn't actually write this section (this section was there before I started editing the article). That is exactly why I kept a cleanup tag (per your reason). I will rewrite it and add it back later. Point-set topologist (talk) 21:16, 19 December 2008 (UTC)

As with groups the symbol ⋅ is usually omitted and multiplication is just denoted by juxtaposition. Also, the standard order of operation rules are used, so that, for example, a + bc is an abbreviation for a + (bc).

As noted below, multiplication in a ring need not be commutative. Some fields such as commutative algebra and algebraic geometry are primarily concerned with commutative rings. Mathematicians writing in those areas (such as Alexander Grothendieck in Éléments de géométrie algébrique) frequently use the word ring to mean "commutative ring" by convention, and not necessarily commutative ring to mean "ring".

• If G is a group and R is a ring, the group ring of G over R is a free module over R having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.
• Ring of dual numbers: Let є be a formal symbol and F a field. The ring of dual numbers, F[є], is defined as F[є] = {a + bє : a, b in F},with the following addition and multiplication:
(a + bє) + (c + dє) = a + c + (b + d
(a + bє)(c + dє) = ac + (ad + bc
Note that є is a zero divisor: є ≠ 0 but є2 = 0.
• Ring of split-complex numbers: z = x + y j , j2 = +1. A ring analogous to the ordinary complex plane but substitutes conjugate hyperbolas for the unit circle.

Some of these examples need to be re-written into their own section. In particular, I think that the last two should be deleted altogether. Point-set topologist (talk) 10:42, 22 December 2008 (UTC)

• If n is an integer, and a an element of the ring define na as one would by viewing a as an element of the additive group of the ring (that is, 0 if n is 0, the sum of n copies of a if n is positive, and the opposite of (−n)a if n is negative.) The integer, n, is usually written for the ring element n1. Then:
• The two definitions of na coincide, that is, first, with n viewed as an integer as above; second, with n meaning the ring element n1 and multiplication in the expression na taking place in the ring. Thus the integer n may be identified with the ring element n. (Except that more than one integer may correspond to a single ring element this way.)
• The ring element n commutes with all other elements of the ring.
• If m and n are integers and a and b are ring elements, then (ma)(nb) = (mn) ⋅ (ab)
• If n is an integer and a is a ring element, then n ⋅ (−a) = −(na)

Above is more of the article which I think should be removed. Point-set topologist (talk) 10:48, 22 December 2008 (UTC)

## Needs motivation for concepts

Point-set topologist (talk) 17:19, 21 December 2008 (UTC)

## Improvements necessary

The following improvements are necessary (in order of priority):

• More inline citations (I don't have access to a library for sometime so if you could add even one citation I would be grateful). Especially, if anyone has the book by Lang, could they find the page numbers corresponding to certain facts in this article (Lang is the most cited book on group (mathematics))?
• History has to be expanded like group (mathematics). Perhaps people could rewrite the text from the MacTutor archive (cited in article).
• The "Examples and applications" section has to be expanded; particularly a summary (like group (mathematics)) is necessary at the top of the section.
• "Group properties of rings" has to be expanded.
• The lede should be expanded.
• "Category theoretical description" should be expanded.
• There needs to be something on Jordan rings

(I have too much on my hands to do all of the above now (in real life!)). Point-set topologist (talk) 11:14, 22 December 2008 (UTC)

The article is a bit too long despite that it doesn't cover many important topics. I propose we delete many discussion on elementary properties and some specific examples. For example, is a Lie ring important enough to mention? (I'd never heard of the term before I came across this article.) -- Taku (talk) 13:41, 22 December 2008 (UTC)

Thanks for working on the article!

Lie rings, Jordan rings, Lie algebras etc... are very important in mathematics. Just because you don't know something doesn't mean that it is not important. For example, do you know what an algebraic variety is (if you do, ignore this)? I didn't even learn that when I knew algebraic topology quite well but yet it is a crucial object in algebraic geometry. I am just saying that there are important concepts in mathematics (that some people do care about) that most of the population does not know (Lie rings however are fairly common).

I don't think that the article is supposed to cover everything (although I could be wrong). Compare it to group (mathematics) which still omits fairly important concepts.

I don't know if deleting some examples is a good idea. As I said, group (mathematics) has a lot of examples. Remember, that not everyone is as clever as you think they are (read the article as if you have never heard of rings before; would you be able to understand it?).

Point-set topologist (talk) 13:54, 22 December 2008 (UTC)

I fear that after less than a week in his new incarnation, Topo's civility may already be degrading. Topo, if you had taken the time to check, you would see that User:TakuyaMurata (who has a master's degree in mathematics) has a wealth of links on his userpage, including a whole list devoted to Lie algebras and several links to algebraic geometry topics. What he was saying is that the name Lie ring was unfamiliar to him.
Anyway, given that we have an article which is not yet scratching the surface of the enormous theory of rings -- i.e., with an associative multiplication -- it is an easy choice to exclude consideration of non-associative algebras. Plclark (talk) 14:26, 22 December 2008 (UTC)

I should have been more specific. (Sometimes, when you are in a grad school, you get the impression that everyone knows Lie algebras.) Yes, my point was the term "Lie algebra" is probably much more known, and it probably makes more sense to discuss Lie ring as one form of generalization. (At least that's how I understand.) In other words, this article shouldn't include the list of definitions of Lie ring, given that this article is about ring in general. (In fact, I just noticed the article doesn't mention noetherian ring, the most important ring!) Since the discussion is too slow, I'm simply going to delete materials that in my opinion are non-essential. (since I'm on Christmas holidays, I should have a plenty of time to devote to this major revision.) -- Taku (talk) 15:21, 22 December 2008 (UTC)

Ok, I put a new version at Ring (mathematics)/Draft. Could you explain what's wrong with it, Point-set topologist? It needs more polishing, but to me it is in a good direction. -- Taku (talk) 17:25, 22 December 2008 (UTC)

User:Point-set topologist told me at my talkpage that he wants to have the organization similar to that of group (mathematics). Since groups and rings are distinct concepts and not even remotely similar (similarity in definitions doesn't correspond to that in properties or behaviors as we all know), I don't see why. -- Taku (talk) 17:37, 22 December 2008 (UTC)
BTW Lie rings get a lot of use in group theory. The idea is to take the success of Lie algebras over fields, and allow it to work over fairly tame rings like Z/nZ, for n not prime ("Lazard correspondence" is one name for this). In general, it is useful to look at a Lie ring over R whenever you look at a Lie group over R, so for instance R could be the integers, the p-adic integers, or various small extensions ("maximal orders" tends to go along with this). For the most part though, Lie rings are considered as Lie algebras over slightly more general rings (some references don't require Lie algebras to be over fields anyways).
I agree that mentioning connections between Lie algebras and rings is appropriate in this article, but that there is no need to specifically go into Lie rings. I would prefer that both ring to Lie algebra and Lie algebra to ring functors were mentioned, as the current (as of a few days ago) discussion was one-sided. JackSchmidt (talk) 17:43, 22 December 2008 (UTC)

Thank for enlightening me. The note on a Lie ring would be much nicer than simply the list of definitions. The current article (new or old) is short of discussion on the use or applications of rings. (And covers too much on elementary examples and properties.) -- Taku (talk) 18:45, 22 December 2008 (UTC)

Yeah, I'd like some more applied examples. Certainly the Lie algebra to ring thing (universal enveloping algebra) is a cool use. Another nice use of rings is: when trying to understand some set of examples, you form a ring generated by that set and study it instead. The representation ring and Burnside ring are reasonable examples of this. From my point of view, you have these modules that you declare to form a basis of the ring, and use direct sum and tensor product as the ring operations. You can extend the "scalars" to include rational numbers to let you only take "half" of a module, and you can extend to complex numbers to allow using more analysis. Actually, just allowing "negative" modules is already helpful, as it is often easier to prove that X-Y is zero than X=Y directly. I think there are lots of other "combinatorial" examples like this (maybe this is called algebraic combinatorics). Probably one could say something about function field as a tool to understand curves and surfaces. I tend to think of cohomology rings as a shortcut to understanding as a whole the cohomologies defined separately in each dimension. JackSchmidt (talk) 19:01, 22 December 2008 (UTC)

Interesting (which is a standard response when you don't know what is being discussed.) Yes, a cohomological point view might be interesting to discuss in the article. Since my expertise is limited to commutative rings and algebras, I'm not a good writer for that task. -- Taku (talk) 20:00, 22 December 2008 (UTC)

## Big changes

Howdy, I noticed the big changes and the two reverts. For the most part Taku's changes looked good, but it did delete an awful lot of material. Unfortunately the revert also deleted Taku's new material. The revert-revert again deleted some of the old good material. I think it would be a good idea to try and merge back the good parts of PST's material deleted by Taku in small changes with edit summaries. For instance, at first glance it appears that Taku deleted some important references that should probably be merged back in somehow (but not by reverting his changes). JackSchmidt (talk) 17:43, 22 December 2008 (UTC)

Yes, sorry about the radical changes. I went more than I initially planned. I deleted lots of materials since I was trying to create a better organization. After the reorganization, I thought we can just readd old materials back. Again, I should have been more clear on this. -- Taku (talk) 17:49, 22 December 2008 (UTC)

Mainly, I would advise Taku to see group (mathematics) (to check the style of the article). The article should be accessible (don't use complex terms in the lede). Please have a look at the link and compare both articles. This should explain why your change weren't completely appropriate. Please ask User:Jakob.scholbach for his opinion on your changes. Point-set topologist (talk) 18:02, 22 December 2008 (UTC)

Like I said above, I don't see why we have to follow the format or structure of group (mathematics). Could you explain why you are unhappy with the changes. Is it only because the style of the new version differs from that of the group article? -- Taku (talk) 18:23, 22 December 2008 (UTC)
OK. I understand that I must explain word by word. Here is the main criticism:
• Lede: The lede has to give a basic overview of the subject. Now, you obviously know that a binary operation combines two elements to form a third but does the general population know this? Would the general population know that addition and multiplication are binary operations and not just addition on the real numbers? I suggested that you read group (mathematics) because that is how the subject should be treated. We should make it understandable to the general population.
• Grammar: I certainly respect the fact that you know more than one language (I myself know more than one language and therefore I know how hard it is to learn another language). However, sometimes, your edits are not gramatically correct (mathematically correct of course!). This means that we (I) have to go through some of your edits which is a (minor) problem. I don't want to be rude; you write English fluently but there is always the minor grammar problem now and then. Please see the criteria for featured articles.
• I spend a lot of hard work to improve the article. Of course I don't own it, but removing almost everything I wrote (some 500 edits) in 10 edits is not something that should be taken lightly. I suggest that we keep the current lede (which beautifully explains the concept of a ring to someone who does not know anything about it) compared to your lede which requires some depth of knowledge to comprehend. One other problem with your lede (you wrote):

Commutative rings, a ring with the property ab = ba, are much better understood than noncommutative ones

"with the property that ab = ba": for which a and b? All a and b in the ring? Mathematical precision is very important especially in featured articles.

These were the main problems with (most of) your edits. I will try to incorporate them into the current article but for now, I have to go. Nevertheless, I appreciate your efforts in this.

Point-set topologist (talk) 19:33, 22 December 2008 (UTC)

(after edic conflict) I think I find the structure of TM's version to be a bit better. Bringing the example of the integers to line one should help the layman as an intro with many new technical terms can be intermediating. I do prefer giving the definition first imediately followed by a worked example rather than vica-versa but PST's description of the properties seems to read better. I never like the show/hide button in articles especially when the they are only hiding one or two lines. I agree with Jack that there are good parts in both versions. --Salix (talk): 19:37, 22 December 2008 (UTC)
Yes, let me emphasize: there are good parts of both versions. I like the structure of TM's version better (and I think that was more or less his main goal), and that using his structure we should merge back in the good parts from PST's version into their new places. It's not a simple copy-paste, as some of the emphasis has changed, and as PST put so much work into it, I thought it was a good idea to let him merge back in his own writing. I would not worry too much about the intro, since it will have to be rewritten as the article progresses. It should summarize the article; it is not an introduction to the awe-inspiring majesty that is the mathematical concept of ring, but rather an introduction to this encyclopedia article which is but a tiny drop in the sea of writings about rings. As the article changes, so must the introduction. JackSchmidt (talk) 19:50, 22 December 2008 (UTC)

Template:Edit-conflict Certainly. TM's version did have some extra information which should be added back. On the other hand, I know that Jack has worked on Group (mathematics) and I was aiming to have exactly the same structure as that. Anyway, I think that the current lede should be kept. I will compare and explain (the following is TM's lead):

In mathematics, a ring is an algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers.

I don't think this sentece is well-written. Notice: "A ring is an algebraic structures". More than the extra "s" at the end of "structure" (grammar), how many people would know what a algebraic structure is? My lede explained this clearly. Point-set topologist (talk) 20:06, 22 December 2008 (UTC)

In technical terms, a ring is an abelian group with multiplication that distributes over addition (i.e., c(a + b) = ca + cb.)

This sounds as if "a ring is an Abelian group with multiplication"; it is under addition. The c(a+b) = ca + cb is not TeXed properly. "Technical" should be "mathematical" etc... Point-set topologist (talk) 20:06, 22 December 2008 (UTC)

Commutative rings, a ring with the property ab = ba, are much better understood than noncommutative ones.

It should be, "Commutative rings (a ring with the property ab = ba), are much etc... Even that is wrong because ab = ba is not TeXed and it should be 'ab = ba for all a and b in the ring (mathematical precision). Point-set topologist (talk) 20:06, 22 December 2008 (UTC)

Due to its intimate connections with algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, their theory, which is considered to be part of commutative algebra and field theory rather than of general ring theory, is quite different in flavour from the theory of their noncommutative counterparts. A fairly recent trend, started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups, attempts to turn the situation around and build the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (non-existent) 'noncommutative spaces'.

This is copied (completely) from ring theory. Not that this is a problem but it is still not written properly. Point-set topologist (talk) 20:06, 22 December 2008 (UTC)

The ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

Throughout the article, a ring is assumed to be unital and associative. See also glossary of ring theory for additional terminology.

What does 'unital' and 'associative' mean (people may not know)? This was (appropriately) placed in "notes on the definition". Point-set topologist (talk) 20:06, 22 December 2008 (UTC)

This is why I think my lede is better. If necessary, I can critize the other content too (there is of course good content and I will add that back when I next get the time to do so). I think with such dramatic changes, it is always good to explain first and then change. Point-set topologist (talk) 20:06, 22 December 2008 (UTC)

Point-set topologist (talk) 20:06, 22 December 2008 (UTC)

Thank you very much for responding to my post. Now I can understand what made you unhappy, (and I'm very happy that I now know it.) To respond,

(i) the lead. This is a delicate issue, since it's about the personal preference. I can see why you prefer your version to mime. I personally try to follow the advice at WP:LEDE, which states the lead must be the summary of the article, not an introduction to the subject. The old version even doesn't discuss commutative and non-commutative rings or applications in algebraic algebra. (I admit the later is related to my interest.) So, I deleted the mention of the closure property, which is rather a technical detail, and I copied-and-pasted one from Ring theory. But like I said it's more about personal tastes, and I'm happy leave the community to decide what type of the lead it wants. (My position on this should be clear.)
(ii) Grammar. Yes, that's a problem :) I was interested in the organization, and so my writing was rather rushed. (I thought there is a plenty of time to polish it later, so I didn't mind. I intended to go over later to fix syntactical errors. or maybe I can't and need others' help on this, though.)
(iii) I'm sorry that if I made you feel like, somehow, I demolished your work. (Sad to say, you just have to be used to that feeling in Wikipedia. Natural selection makes Wikipedia better.) My intention was to tighten and short the article to create rooms for more discussion on examples, application and such. The article has to include core materials that can be found in any standard algebra textbooks, and the old version, (believe me, which lacks a lot of materials on core topics), simply didn't have the rooms for such additional content.

Let me thank you again for letting me know why you didn't like the new version, now we can go forward. -- Taku (talk) 19:55, 22 December 2008 (UTC)

Thanks for the understanding. I know that you are certainly an expert on the subject. More important than re-organization would be if you perhaps write about the important concepts on the topic and then add it to the article.

Thanks!

Point-set topologist (talk) 20:08, 22 December 2008 (UTC)

Let us now focus on Plclark's suggestions (thanks Pete, I greatly appreciate your input). Point-set topologist (talk) 20:51, 22 December 2008 (UTC)
But that doesn't work for the reasons spelled out above; whence, the reorganization. (I'm really not an expert on algebra since my background is an analysis.) Do you think you can work with my version? by fixing errors and polishing the text? -- Taku (talk) 20:49, 22 December 2008 (UTC)
I will look at them now. But as I explained I think that my lede is better (so I may not add back your lede but I will add back your other edits). Point-set topologist (talk) 20:51, 22 December 2008 (UTC)
I don't care much about the lead. Do you think you can accept the "basic structure" of my version? If not, we have to submit ourselves to the community and let them decide which version they like. Like I said above, the structure of the old article is unacceptable to me (whence, the change). Nothing personal, but that's just, well, preference, I guess. -- Taku (talk) 20:54, 22 December 2008 (UTC)
Looking over your changes, I must say that some of the content you have added is important. However, I encourage you to read WP:MOS; it is important that you not only contribute content but make sure that you don't overlink, use "we" (unencyclopedic), and destroy structure etc... Point-set topologist (talk) 20:59, 22 December 2008 (UTC)
I will add back your content now since you insist. But I strongly advise that you read group (mathematics) and note the structure. The structure should first start with a familiar example (not with the formal definition) and then explain how this generalizes. As I said, Jakob is an expert on FA's and GA's (and if he agrees with your structure, fine). Point-set topologist (talk) 20:59, 22 December 2008 (UTC)

You didn't answer my question. -- Taku (talk) 21:07, 22 December 2008 (UTC)

I don't agree with the structure. As I say, please consult Jakob (an expert in writing Wiki articles). I don't really think that your changes were for the better. Point-set topologist (talk) 21:18, 22 December 2008 (UTC)

Ok, then we seek inputs from other contributors.

This might be to beat the dead horse, but about FA and group, I don't see why we have to follow the structure of the group article. (I personally see it is a bad example.) So, some explanation on my intention behind the radical changes might help. Wikipedia being a reference work, it is important to get to the point as quickly as possible. This is why I moved the definition above some motivational example. While not every reader knows rings, not every reader is non-mathematician either. Every article in Wikipedia has to be organized so that readers can find materials on topics that interest them. When reorganizing the article, I was able to see the efforts to increase the accessibility, which is fine, but that shouldn't conflict with the efficiency with the introduction of materials. This is why many math-related articles are accused of being terse, which is justifiably so but you must understand there are good intentions for that format too. While motivations are important, we cannot forget Wikipedia articles are meant for reference. -- Taku (talk) 21:25, 22 December 2008 (UTC)

Group (mathematics) has been accepted (by the community) as one of the best articles in Wikipedia (that is why it is FA). If you think otherwise, the community, probably won’t agree with you. Point-set topologist (talk) 22:00, 22 December 2008 (UTC)

But that narrow "community" doesn't include me, and doesn't represent the whole community of Wikipedia. Anyway, I have to be more specific. What I meant to say is that that article is a bad example "for this article". Since a ring has much more complex structure than a group, it makes sense to employ a different format. Don't you agree with that? -- Taku (talk) 22:02, 22 December 2008 (UTC)
edit conflict: ::see democracy. The "community" includes math administrators (power). —Preceding unsigned comment added by Point-set topologist (talkcontribs) 22:08, 22 December 2008 (UTC)

And you think admins are the power that be? That only proves how you're new to Wikipedia. -- Taku (talk) 22:13, 22 December 2008 (UTC)

## Outline

I think using the group (mathematics) outline was a very wise thing to do while PST was adding all that new content. Many wikipedia articles have basically no organization at all, and using an FA as a model is a very good idea. However, now that PST has added all this good content, it has become clear that the article needed to be reorganized. TM's new organization is not perfect (and leaves out some pretty huge deals from my point of view), but I think it is good for this stage of the article writing. Once we get more of the material added (much of which PST already asked for help adding), the article will probably need another rebalancing. Right now I think the consensus (Salix, myself, and the other leading JS) is that TM's draft should be used for its organization, and that we should merge back in the good content from PST. Once the outline is settled, I'll be more willing to add my content to the article rather than the talk page. JackSchmidt (talk) 23:00, 22 December 2008 (UTC)

Hi Jack,

I explained why I don't agree with TM's ordering on your talk page, but my computer has serious problems and this did not get saved. But mainly, TM's ordering was not accessible; it said in the lede that "all rings in this article are assumed to be unital and associative" before explaining the meaning of these terms. In general, mathematical terms are written in the section before they are explained. But, the ordering at the moment is low priority. It is more important that we add new content to the article. In particular, Plclark has given some suggestions for improvement and we could probably work on that (if the consensus agrees later, we can change the ordering; there is no point in re-ordering the article every few days).

Thanks!

Point-set topologist

Other problems with TM's version: Use of "we", grammar, mathematical precision, accessibility etc... (One reason for these problems is that he copied a lot of material from other Wiki stubs). —Preceding unsigned comment added by Point-set topologist (talkcontribs) 18:24, 23 December 2008 (UTC)

Copied from Taku's talk page:

Taku, I would like to emphasize the comparison of the two ledes. First of all, my lede is a summary of the article; it discusses (in simple terms) what a ring is. Your lede on the other hand, is definitely not a summary of the article. It gives the applications of rings to other branches of mathematics. If you want to add back your lede, I advise you to add it to the "examples and applications" section (and again I would like to request you to read my criticizm of your lede on the talk page of ring (mathematics)).

The other problems with your version is that you seem to have blindly copied text from other articles. While this is not against any Wikipedia policies, there are many errors in those articles. So I suggest that in future, before copying and pasting, check for errors first (and this will also ensure that you won't get the blame for those mistakes).

Also, as you confirmed, you intended to fix up your version for grammatical mistakes later. How much later? Fixing up those grammatical (and mathematical errors) would take months unless you want to work the whole day (for 5 days like I have done). I intend to get this article to GA as soon as possible (don't postpone).

The other thing you said was that the choice of version depends on the person. It doesn't. It only depends on the Wiki policies. Some criticizm of your version is that you use "we" too often, your version is not accessible (i.e don't use mathematical terms before you actually explain their meaning; I suggested you to see group (mathematics) because there everything is explained clearly).

I want to put this in the politest possible way, but your version is start class. It requires significant revision (which I can't do all over again (it took a lot of hard work)) to make good. Technically, the other Wikipedians say that your ordering is better. I would appreciate it if you understood my point of view (then I can understand yours).

Thanks!

Point-set topologist —Preceding unsigned comment added by Point-set topologist (talkcontribs) 20:03, 23 December 2008 (UTC)

I also want to advise Taku to work on this article (not major edits) but just corrections and such. I have already explained what is wrong with Taku's version (except for the order of material, I think most people agree that his version is not appropriate, right?).

PST

First, about the lede. I believe the lede should not only introduce a subject but is the summary of the article. From this follows inevitably that the lede may contain many technical terms and mentions of important examples and applications, which may require the readers to have some prior background. We don't make the intro easier to understand so that a greater number of people can understand it. No, we make the intro as informative as possible to maximize the value of the article for the reference purpose. In other words, the lede should be much more than discussing what a ring is in laymen's terms. Hence, my changes. (In fact, many math-related articles follow this style, I believe.)
Yes, but is your lede a summary of the article? Your lede comprises of two opening sentences: both mathematically and grammatically incorrect. My lede explains those first two sentences in depth. Second of all, the remaining part of your lede does not at all summarize the article. It just explains the applications of ring theory to other parts of mathematics (and science). Remember, this article is about rings; not ring theory. Your lede would have been much more appropriate for ring theory (once corrected) because there it would summarize the article.
Also, I don't understand how you got the idea that my lede is in laymen's terms. To write a good lede, one must make it as accessible as possible as well as mathematically precise. My lede gives a mathematical term, then explains it. Don't you think that this is better than writing just the mathematical term. Furthermore, even though my lede does not use complex mathematical terms, at least it is correct, yes? Your lede on the other hand is both mathematically and grammatically incorrect and is better suited to the article, ring theory. In fact, this is not your lede (which is a good thing). It is copied and pasted from another article (start class) and hence the errors. I suggest, that if you really want your lede, add it to beginning of the "examples and applications" section. But first, I strongly suggest that you rewrite it in your own words (with corrections). --Point-set topologist (talk) 10:31, 24 December 2008 (UTC)
Second, about GA. Why hurry? I think it is important to focus on the long-term goals (i.e., increasing and improving the coverage of math-related topics) than short-term ones (i.e., GA or FA status). Or, at least, GA or FA statuses are just not my priorities. To me, informative articles beat articles with grammatical errors. So, I tend to prioritize the former. I understand that some people have other priorities, but, to me, that just misses the point entirely. GA and FA statuses are meant to give some quantitate information about what is done and what is not done. They are not something to be sought after. (Students study for exams, but ideally students should study for the sake of the subject itself and use exams to help them access their degree of understanding.)
Thirdly, about policies. I don't think you understand how Wikipedia works. Policies and guidelines are meant to make contributors productive (not put constrains to them). The point is that, without codified sets of rules, contributors have to spend too much time on matters that have come up over and over again in the past. (For example, the notability policy doesn't dictate what can be included to Wikipedia; that's what AfD is for. But with the policy we can save a lot of time not doing actual AfDs.) In other words, citing the policy is never an argument, and, when there is a disagreement, you have to make a case yourself. This leads to my
Forth point. Wikipedia being a reference work, its articles are meant to be used for the reference purpose. It follows that, for one thing, you shouldn't expect your readers to read the article from top to bottom. You can't even expect they read a section thoroughly. Some simply search for a keyword, and read only the sentence that contains the keyword. Wikipedia articles have been optimized for this usage. In other words, we (often) don't employ a textbook format where a motivational example is followed by a formal definition and discussion on how early examples are connected to new ideas. That's a very beautiful format, but that's not how Wikipedia articles are usually written because of the aforementioned usage pattern. It is much easier for me to show this than explaining it in words: my draft speaks for itself. In fact, this is related to the lede issue. It is important to state the convention in the intro, since readers (and in fact other contributors) who already know rings probably don't read the definition or examples sections, they would be lost as to what convention is used for the article.
Time and time again, I have requested you to read group (mathematics). Group (mathematics) presents the topic in an easy and accessible manner, well written, and provides information. As you say, you are not interested in the FA procedures. Well I am and that was my goal when I improved these articles. If you don't like FA procedures, I suggest that you improve other articles which are not close to FA. When articles reach a good quality, it is time for FA.
I also intended to add more information which I would have done instead of wasting my time typing here. My point is that your version was done without discussion, with your purpose (not to make the article FA, but make it back to how you like it), and in your style (grammatically and mathematically incorrect). I don't mean to sound rude but I have kept calm since I saw your changes. I approached it in a logical manner (rather than getting angry). Before my changes, the article was start class. The length of the talk page was five times the length of the article. What I thought was: why are people wasting their time talking instead of implementing? But again, it seems that we have gone back to talking. --Point-set topologist (talk) 10:31, 24 December 2008 (UTC)

(P.S. I'm not personally offended at all if that's what worries you. It's been long since I started the contribution to Wikipedia, so I'm used to the dispute. Actually, I was very surprised by how my edit upset you, for that, I'm very sorry. I don't even insist that we use my version, since now I know how this article meant for you and my version doesn't mean (personally) anything to me. One good editor is much more valuable than one good article.) -- Taku (talk) 22:59, 23 December 2008 (UTC)

As I said, I will add the good parts of your version; I won't add back what you have re-written nor will I delete the content that you have deleted. What was most upsetting to me (I was not actually upset) was that you made these deletions and re-writes without discussion. In future, I suggest, that you discuss before you make such major edits.

--Point-set topologist (talk) 10:31, 24 December 2008 (UTC)

How many times do I have to tell you? I have politely discussed with you, agreed to add your changes and yet you persistently insist that your version should be chosen. You say that Wikipedia articles are work in progress. What you fail to understand is that your version will take at least another 500 edits to improve. I have already worked hard on this version and yet you want to undo it with 10 edits. Please stop discussing here and edit another article. If you do not wish to do this, I won't respond to any further comments. --Point-set topologist (talk) 16:14, 24 December 2008 (UTC)

## Image of text?

Why does the article begin with an image of text? This is frowned upon (see Wikipedia:MOS#Images_as_text), and probably won't make it through the GA/FA filter. That image could easily be duplicated as a wiki table, and perhaps a better (pictorial) image could be found (though I don't have any ideas). Staecker (talk) 12:55, 23 December 2008 (UTC)

Well somebody moved it down (which I like), but it's still an image of text. I've created and inserted {{Algebraic structures}}. Feel free to play around with it (isn't it nice that you can edit it easily? (unlike the image)). Staecker (talk) 12:56, 24 December 2008 (UTC)

Thanks for the image! It looks much better. But I think we need something more visual at the top. --Point-set topologist (talk) 13:42, 24 December 2008 (UTC)

## New section: some examples of the ubiquity of rings

I have decided to spruce up the article by adding some higher-level content. I made a new section with the above title. The idea of this section is to discuss some of the many ways that one can (functorially, although how explicit this should be is as yet unclear to me) associate a ring to some other kind of mathematical object. I think this ia good way to see that rings are interesting and useful in many different areas of mathematics. Here are some ideas for such things: others should, as always, feel free to add more:

• tensor/symmetric/exterior algebra of a module
• de Rham cohomology of a manifold
• Boolean ring of clopen sets of a topological space
• universal enveloping algebra of a Lie algebra
• ring of continuous functions vanishing at infinity of a locally compact Hausdorff space
• Witt ring of a field
• ring completion of a commutative semiring
• K_0 of a commutative ring R
• K_0 of a topological space
• coordinate ring of an affine algebraic variety
• homogeneous coordinate ring of a projective variety
• Chow ring of a variety
• semigroup ring k[M] of a semigroup M (over a base field, or ring, k)
• representation ring of a finite group (mentioned above, I believe)
• Clifford algebra of a quadratic form

And so forth. Plclark (talk) 13:00, 24 December 2008 (UTC)

Great work! Now that all the basic content is added (like morphisms, subrings etc...), we can start giving summaries of the higher level mathematics of rings. What about something on schemes and spectra? I think I could write that a bit later (if no-one does). Also, thanks for re-writing the notes. I think it now gives some useful information to the intermediate reader. I will patch that section up (in terms of linking, for instance I don't think that equivalence relation is linked) now. —Preceding unsigned comment added by Point-set topologist (talkcontribs) 13:40, 24 December 2008 (UTC)

I strongly disagree. This article shouldn't be just about "introduction to ring theory", it should be much more. By this I don't mean to suggest the article discuss ringed spaces or, even more generally, scheme theory, because they're topics in commutative algebra. Ditto to things like Nakayama's Lemma and Hilbert XXX theorems. (Maybe I'm only ignorant, but I don't know any non-commutative counterparts of them.) The article should be about main ideas and procedures (e.g., various constructions) that are applicable to even to non-commutative rings. On the other hand, though my background isn't on non-commutative rings (not even algebra), I believe, for example, localization is still an important tool for non-commutative rings. (e.g., microlocalization) And I don't even have to mention numerous applications to non-commutative geometry. The main focus of the article should be on such topics. In particular, it follows from this view that the article shouldn't contain many examples of commutative cases; they belong to commutative ring, and that examples should focus more on non-commutative cases (e.g., matrices), since we don't have noncommutative ring (should we create one?). (I guess this explains why Point-set topologist can't accept my proposal for the structural changes, at least partly.) -- Taku (talk) 15:28, 24 December 2008 (UTC)

I have a few points to mention:

a) There are articles on every concept in ring theory. If there is already an article, why add more here? This should give a basic background and then link to the articles which explain the "advanced concepts" in depth. Plclark has done this and I agree.

b) Secondly, I was not rude to remove your link. I was as polite as possible when I saw your changes not to mention the fact that I kept my cool. If you like your version, go ahead and do it to group (mathematics). I think that if you do, God knows how many people will revert your changes. In this case, people don't say anything because it is not they who have done the hard work.

Therefore, I am removing your link (you specifically stated that you agree with my version), and am going to edit the article. If you want to make an article in your style, I suggest you do it to another article or use the sandbox. As I said, go ahead and make your changes to group (mathematics) and see what other people say. Please stop wasting time here (and other's too; we want to expand the article, not gabble on here as people have done for the past 6 years). --Point-set topologist (talk) 16:09, 24 December 2008 (UTC)

Apparently I have to criticize your new lede (per request):

In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers.

Gives an example of the integers too quickly. The first sentence should start by writing about binary operations. It is not clear to the reader yet whether addition is a binary operation or not (if someone reads this article, he will know very little about rings (if at all anything)). --Point-set topologist (talk) 16:31, 24 December 2008 (UTC)

In technical terms, a ring is an abelian group with multiplication that distributes over addition (i.e., c(a + b) = ca + cb and (a + b)c = ac + bc.)

This is completely incorrect. For a start, the sentence sounds like "a ring is an Abelian group with multiplication". No. It is an Abelian group under addition. Even then, it is not clearly stated what a ring is under multiplication (a monoid). --Point-set topologist (talk) 16:31, 24 December 2008 (UTC)

A ring is one of the two main subjects of study in elementary abstract algebra, the other being a group. Examples of rings include: the set of all integers, the space of continuous functions, polynomial rings and the ring of matrices (but not the set of traceless matrices).

"Examples of rings include: the set of all integers, the space of continuous functions ...". First of all, the set of all integers is not a ring unless you impose the necessary binary operations. Second of all, "the space of continuous functions", has no meaning because you are not specifying their domain and range (the range has to be a topological ring). I don't know head or tail of what you mean by the "ring of matrices" (mathematical precision again, you have to specify the field etc...). --Point-set topologist (talk) 16:31, 24 December 2008 (UTC)

Commutative rings (rings with the property ${\displaystyle ab=ba}$) are much better understood than noncommutative ones.

It is not understood that a and b are in the ring. I have no idea head or tail of why commutative rings are much better understood that non-commutative ones (I know but you have to explain this to other readers). --Point-set topologist (talk) 16:31, 24 December 2008 (UTC)

Due to its intimate connections with algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, their theory, which is considered to be part of commutative algebra and field theory rather than of general ring theory, is quite different in flavour from the theory of their noncommutative counterparts. A fairly recent trend, started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups, attempts to turn the situation around and build the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (non-existent) 'noncommutative spaces'.

This paragraph does not make sense (if you ask me, it should be removed; stop copying from other articles). --Point-set topologist (talk) 16:31, 24 December 2008 (UTC)

The ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

"The ring theory". What sort of language is that? This article is about rings, not ring theory. Write this lede at ring theory as I have requested you several times. --Point-set topologist (talk) 16:31, 24 December 2008 (UTC)

The word ring comes from the German word 'Zahlring', meaning number ring. Although the first definition of rings was given by Fraenkel (1914), it was Hilbert who popularized the term.[1] The term also appears in measure theory: rings of sets. They are not considered a ring in the sense used in algebra.

Sigma rings are not rings in the algebraic sense. Why mention sigma rings here anyway? --Point-set topologist (talk) 16:31, 24 December 2008 (UTC)

Throughout the article, a ring is assumed to be unital and associative. (See nonassociative ring for nonassociative cases; in particular, Lie ring.) See also the glossary of ring theory for additional terminology.

You have not explained what unital and associative mean. This should be given after the formal definition (so you have not ordered the article correctly). --Point-set topologist (talk) 16:31, 24 December 2008 (UTC)

I don't want to waste anymore time here. Again, I request you to edit another article; if not, I will not respond to anymore of your comments (sorry but I have been as polite as possible and yet to continue to behave in this (impolite) manner). I strongly recommend that you drop your version now. --Point-set topologist (talk) 16:31, 24 December 2008 (UTC)

(I don't respond to your argument as requested.) At least you should acknowledge the disagreement. Why is that so hard for you to accept that someone might not agree with you? (So, he or she is going to change an article accordingly.) Like said above, that's how Wikipedia works; either you accept this nature of the editing model or you must leave. This isn't your article; you don't have the final say to the format or style of the article or what can be included or not. (You are apparently acting that way.) You are the only who is being too stubborn. Like said before, I'm not planning to replace the article by my version, but you should at least accept the existence of the disagreement. -- Taku (talk) 16:55, 24 December 2008 (UTC)
I accept the disagreement and have accordingly promised to add parts of your version back. The point is that you say that you don't want your version, then that you do etc... I have explained why your lede is not appropriate and you have not responded. I appreciate your efforts in this, but I would like to request you to allow us to get this verson to FA. So equivalently, I am saying that I won't respond to anymore of your comments: I can't prevent you from writing stuff on the talk page but at least I can do this. --Point-set topologist (talk) 18:07, 24 December 2008 (UTC)

Copied from my talk page:

### A lurker's perspective on ring (mathematics)

May I politely suggest, PST, that you chill out and assume good faith? To suggest that Taku is simply wasting his time and others' is fairly ridiculous. To ask him to stop contributing to this article and discussion is also inappropriate. You may want to check out WP:OWN. I realize (and appreciate) that you've put a lot of work into your version, but that alone doesn't make it worth preserving. You having "kept your cool" is good, I suppose, but I see absolutely nothing that anybody's done here which should cause you to lose your cool in the first place. People changing around and deleting your text is not rude-- it is exactly what happens on Wikipedia all the time. Attacking Taku repeatedly for advocating his version (exactly what you are doing for your version) is unlikely to win you any allies. (I'll watch here if you want to reply, but I'm not really interested in being drawn into an argument- consider this my (unsolicited) two cents.) Staecker (talk) 16:40, 24 December 2008 (UTC)

I will chill out. I did get a bit upset but the fact is that I explain what is wrong with his version, he does not respond to this, and again says his version is good. This takes up a lot of time for me. I am sorry for getting upset and I guess this was unnecessary. People changing around and deleting my text is of course what I want: I want people to improve the article. I was upset with Taku because he deleted half the article without discussion and then re-wrote it in his own words. I will continue to keep my cool and once again I am sorry about getting angry (perhaps you might have felt the same way if you were in my position).

P.S The other thing that you have to remember is that I am a new user...

Thanks!

PST —Preceding unsigned comment added by Point-set topologist (talkcontribs) 16:53, 24 December 2008 (UTC)

I'm sorry, but what the hell is going on with the page now? There's a mammoth amount of talk about the article pretty much just from two editors i.e. a dispute... is that resolved now (as the below section indicates? If so, what was the resolution? The whole debacle makes it very confusing to everyone else, I think! SetaLyas (talk) 10:58, 28 December 2008 (UTC)

## Recent disputes

I just want to point out that I am sorry for getting upset in the end of the last dispute. In the beginning, I acknowledged Taku's mathematical knowledge as well as his work. But it got to the point where he just ignored my comments. I think that the dispute is now settled and we can get back to editing the article (my apologies again to Taku). --Point-set topologist (talk) 19:58, 24 December 2008 (UTC)

## Concepts required

There needs to be a section on the field of fractions. If anyone has the time, could they please create it (my hands are full at the moment but if no one creates the section, I will do it myself). --PST 17:05, 18 January 2009 (UTC)

Good idea, in fact would it even justify an article? I will think about possibly writing it, whether section or article. JamesBWatson (talk) 11:08, 19 February 2009 (UTC)

You could call it Field of fractions. Algebraist 11:26, 19 February 2009 (UTC)

## GA Review

Article  · Article talk (edit | history) · Watch

GA review (see here for criteria)
1. It is reasonably well written.
a (prose): b (MoS):
2. It is factually accurate and verifiable.
a (references): b (citations to reliable sources): c (OR):
3. It is broad in its coverage.
a (major aspects): b (focused):
4. It follows the neutral point of view policy.
a (fair representation): b (all significant views):
5. It is stable.
6. It contains images, where possible, to illustrate the topic.
a (tagged and captioned): b lack of images (does not in itself exclude GA): c (non-free images have fair use rationales):
7. Overall:
a Pass/Fail:

--PST 17:07, 26 January 2009 (UTC)

Please review. --PST 07:15, 19 February 2009 (UTC)

## Why "ring"?

I always like to know the rationale for mathematical terms; it helps me feel like I know what I'm talking about. What did Hilbert have in mind when he called these things "rings"? I can see why someone would call SO(2) or even SO(3) a ring, but those are groups, not rings. —Ben FrantzDale (talk) 02:09, 28 January 2009 (UTC)

This was asked at the reference desk a while ago, but the only definite answer was from a very dubious source (at least one of the etymologies given on that page is totally wrong). Algebraist 02:20, 28 January 2009 (UTC)
I'll be bold and note that the etimology is vague.—Ben FrantzDale (talk) 22:16, 29 January 2009 (UTC)
A ring is different than its group of units. The non-units form a mathematical singularity within the ring, and the emptyness then suggests the solid torus or anchor ring kind of figure, where the whole is formed around a certain gap. This figure contrasts with a field where a mere point marks singularity. The question is a common one, well-puzzled, and this answer is speculative.Rgdboer (talk) 00:56, 1 February 2009 (UTC)

## Out of place sentence in "Notes on Definition" section

The sentence "The term rng (jocular; ring without the multiplicative identity) is also used for such rings." should be placed after the description of rings without a unit, not after the description of unitary rings. —Preceding unsigned comment added by 4.240.213.232 (talk) 06:16, 19 February 2009 (UTC)

A sequence of revisions has recently been undertaken by Point-set_topologist. Together these have substantially transformed the article, without any discussion or opinions about the specific changes made. I find the original version much clearer and more straightforward to follow than the final version. Does anyone else wish to express an opinion on the matter? JamesBWatson (talk) 11:17, 5 May 2009 (UTC)

Let me also add that clarity of an article, and the extent to which it is straightfoward are not the only aspects that are to be taken into account. For instance, there are many students who may wish to learn the applications of ring theory within mathematics. We must appeal to those students also, as well as laymen. Over the next month, I intend to substantially change the article, in this respect; while keeping the article accessible, I intend to give a full discussion of ring theory and its applications within mathematics. Currently, the topics included, are not the main crux of ring theory, although they may be simple for laymen to follow. --PST 23:50, 5 May 2009 (UTC)
PST: I had read all of the lead, but I confess I had only skimmed through it: if I find time I will read it more thoughtfully, and see if I can come up with some more constructive comments. Perhaps also I should have made it clear that my comment related only to your latest round of changes: there is no doubt at all that, if I consider the total effect of all the work you have done then it is an excellent piece of work. JamesBWatson (talk) 12:55, 6 May 2009 (UTC)

## Article is not encyclopedic

The article in its present state is not encyclopedic. It is too verbose (over 70KB!), with explanations that would be excessive even in a textbook -- which WP is *not* meant to be. It discusses at length topics that are (or should be) discussed in separate articles. The lead section is overly long; while it pretends to be addressed to the layman, it never explains what a ring is, and throws a lot of specailized terms that even a mathematically educated reader cannot be expected to understand.
Repeat, WP is not a textbook. Yes, articles should try to be accessible to the general public; but they should also be as concise as possible, as befits a reference work. And they are not supposed to be self-contained: that is what wikilinks are for. There is no point, for example, in explaining what a binary operation is: readers who do not know that much will never understand what a ring is.
I have tried to trim some of the fat, but there is a lot more left. All the best, --Jorge Stolfi (talk) 02:24, 4 August 2009 (UTC)

I certainly agree that the article was too long. However, the lead of an article is aimed for the widest possible audience. Therefore, although we aim to explain the concept of a ring to laymen, we also aim to describe how the notion of a ring applies to other areas of mathematics. Although I do not see any anything wrong with your edits, I do find that the original lead was better than the current one, in terms of appealing to a wide audience. While we try to make articles as accessible as possible in WP, they are not exclusively for "dummies". Therefore, I feel obliged to revert at least the changes you have made to the lead. If you feel otherwise, we can discuss it here and I will wait for your response before reverting. --PST 05:02, 4 August 2009 (UTC)
Let me give one example. In the old lead, there was some description as to why non-commutative ring theory is almost as active an area of research as commutative ring theory. The current lead fails to note this. To prove my assertion, see P. M. Cohn (a famous non-commutative ring theorist) and note the famous book Non-commutative rings by I. N. Herstein. This is just one example of how some important considerations in the subject are neglected in the lead. --PST 05:06, 4 August 2009 (UTC)
I do not agree that the original lead was better for the lay reader. For one thing, it did not really define a ring, not even vaguely; so all the discussion about the merits and applications of rings will be incomprehensible to readers who do not already know what a ring is.
I have no problem with adding a bit more detail to the lead, such as a note about non-comm. rings being a topic of research. However, there is already a section where the research topics are dicussed in full, and any longer discussion should go there. The lead is not meant to say *everything* that is important; it should only give a *short* overview of the article's topic. Also, any technical discussions about the properties of non-commutative rings (or any special kind of ring) should be moved, whenever possible, to the corresponding article. All the best, --Jorge Stolfi (talk) 07:36, 4 August 2009 (UTC)
Please discuss every change you make. Your recent deletion of the section on "non-commutative rings" was inappropriate. Although I do not think you have bad intentions, I should politely request that you explain dubious changes for otherwise there is no reason why I should not revert. --PST 09:59, 4 August 2009 (UTC)
Let me make it clear that I appreciate your interest in the article. I also agree that many of your changes were appropriate and so should remain. However, I intend to correct some minor points in your changes. For instance, under the section "some important concepts", the concepts of a homomorphism and a subring are included but there is no comment on ideals. In my opinion, ideals are perhaps the most important concept in ring theory (more important than "zero divisor", certainly). Also, "non-commutative rings" do not merely constitute a "research interest". They are of fundamental importance due to their natural occurence in mathematics. Please understand that although I appreciate your edits, some of them are incorrect. --PST 10:18, 4 August 2009 (UTC)
Another example: it is written that "the integer elements of any ring with unity element is isomorphic to the ring of integers". What if the ring is finite or has non-zero characteristic (the statement is almost correct but too basic to be included in its correct version)?
Another example: 1 is not the only unit in the ring of all integers: -1 is also a unit. --PST 11:02, 4 August 2009 (UTC)

I have a few more comments to make. First and foremost, let me note that the article's size has now been reduced to 61 kilobites (a 9 kilobite improvement). On the other hand, note that group (mathematics) is 90 kilobites long and is a featured article. Therefore, although some of the material that Jorge has removed was inappropriate for this article (Jorge's removal was appropriate), much more important material is to be added in the future (notable is a more involved discussion of fields). Therefore, the article's size may well reach 90 kilobites or so in the future. Many of Jorge's changes were appropriate but some were not (in my opininion). Therefore, I also request that Jorge explains big changes to the article before implementing them (small changes are not a problem). --PST 10:52, 4 August 2009 (UTC)

Correction: the initial size of the article was 76 kilobites, so a 15 kilobite improvement is quite reasonable. This would not be without Jorge's input so let me thank him for that. --PST 10:58, 4 August 2009 (UTC)
First, my apologies for the errors that I introduced. I will fix those that I can recognize, ad ask that you help me find the others.
As for the "non-commutative" section, it was moved to a separate article, where one can discuss with more depth and detail. We could copy back some of that material to this article; but, in my opinion, only if it connects to other parts of this article. (For example, subrings and isomorphisms must be defined here because they are used in other important sections, especially in examples; while a single line and a link is probably sufficient for, say, tensor products.)
As for article size, I agree that it is not a problem per se; there is much material that could be added. (For example, what do you think of merging ring theory into this article?) All the best, --Jorge Stolfi (talk) 22:39, 4 August 2009 (UTC)
I might be mistaken but ring theory is supposed to cover a more "advanced" treatment of rings. This article on the other hand is supposed to cover the more basic concepts of ring theory (compare to Group (mathematics) and Group theory). That said, it is practically impossible to cover every concept in ring theory; some concepts must be given more priority than others.
Let me explain my concerns for removing the section on "non-commutative rings". Analogous to finite group theory (where non-abelian finite groups are the most interesting), non-commutative ring theory is also very interesting. Commutative ring theory leads to fields such as algebraic number theory and algebraic geometry which have different goals to ring theory. Therefore, in my view, non-commutative ring theory is more shifted towards the theory of rings (non-commutative division rings alone are complex in nature - there exist non-commutative division rings which are not isomorphic to their opposite ring). Examples in this theory include the famous Jacobson density theorem (first noted in the famous paper by Nathan Jacobson - "On the structure theory of rings without finiteness assumptions") and the Jacobson radical. Other important concepts include also, the representation theory of groups and the Brauer group. Therefore, since non-commutative rings constitute a vast area of ring theory, it makes sense to have a detailed discussion of their theory in this article (the current discussion is not even satisfactory).
Lastly, I agree with you that there are many concepts in this article which are not interesting enough. It is difficult to decide whether concepts such as "the tensor product of rings" or "the direct product of rings" should be discussed unless they lead to interesting consequences (in the context of finite groups, these concepts lead to, of course, "the fundamental theorem of finite abelian groups). With regards to "isomorphism", I do not think much more can be said apart from that it is a "homomorphism which is both an epimorphism and a monomorphism". One could of course mention analogues of the isomorphism theorems for groups (or mention Shur's lemma) but I feel that this is redundant. Similarly, the concept of a subring, in my opinion, is not as interesting as the concept of an ideal (although it is interesting). More emphasis should be given to ideals before a full discussion of subrings is commenced. If you feel otherwise on any point I have mentioned, please feel free to comment. --PST 00:33, 5 August 2009 (UTC)
Dear "Point-set topologist", i am really disppointed by your wholesale revert of my edits. Can't you really see the problems with your version? For starters, the lead section *must* begin with a definition of the article's topic --- succint, but as accurate as possible, and sufficient to distinguish it from other related topics. The present version does not define a ring, not even vaguely. After reading two or three screenfuls of dense prose, the reader has still not been given that basic information. He will not be able to tell the difference between a ring and any other algebraic structure. And, therefore, all that prose will not make any sense to him.
Moreover, the lead section is not the place to go into lengthy explanations as to why rings were invented. At most, there should be one paragraph mentioning the topic's importance and the general are of applications. Detailed discussions belong in the article body, if at all.
Finally, an encyclopedia article should not be written as a textbook. This is not the place to give proofs of theorems, derivations of formulas, or explanations for straightforward examples. Just saying that the integers are a ring is more than enough. Moreover, unlike a textbook, examples (when really needed) should be given *after* the definitions, not before them.
Finally, unlike a textbook or popular science article, a WP article is not supposed to be self-contained and exhaustive. On the contrary, one should avoid as much as possible repeating text from other articles. Non-commutative rings, for example, shold be discussed in their article (did you notice that it now exists?). Here it suffices to say that they are more interesting and more difficult, and give a link to that article. The reader who cares has only to click the link; the reader who doesn't care will be spared the trouble of skipping over a redundant section. Not to mention that the editors' work is more effective if each topic is discussed in only one article.
And so on. Please reconsider. All the best, --Jorge Stolfi (talk) 00:54, 5 August 2009 (UTC)
Dear "Point-set topologist", to answer your comments: subrings should be properly defined here not because they are important, but only because the reader will not be able to understand other examples and definitions without understanding the definition of subring. Ditto for pseudo-ring: the concept must be defined here only for the sake of the "Alternative definitions" section. Ideals are important enough to deserve a section; but it suffices to give a *brief* definition, a couple of examples (with no discussion), and a paragraph or two, at most, *summarizing* their importance and applications. Anything beyond that belongs to their own article. All the best, --Jorge Stolfi (talk) 01:08, 5 August 2009 (UTC)
Sorry for the delay in responding - I did not see your comment until about a few minutes ago. Firstly, I did not delibrately revert your edits in their entirity (a simple undo button would have done that, but notice how several edits on my part have occurred after your edits: not just one "undo"). I attempted to revert those changes which I felt were inappropriate. That said, not all of your changes were inappropriate. I will now attempt to respond to each comment you have made.
With regards to the lead, I do not see why the definition should be included at the start. Leads in Wikipedia articles are designed to summarize various important aspects of ring theory. The formal definition has its place right after the lead, and the reader need only scroll down to read it. Nevertheless, I have now added a link to the formal definition at the end of the first sentence. Should the reader wish to see it, he need only click the link.
Lastly, I urge you to see group (mathematics) in its entirity. This link will address your remaining concerns since group (mathematics) is a featured article that has similar structure ("textbook examples" and "introduction") to this article. Regards, --PST 06:49, 5 August 2009 (UTC)

## Cut too deep

On the 20th of June user Charvest started a section on finite rings that I expanded. A month later it seemed this section was due its own page, so I moved the section to Finite ring. The concern is that active editing in a section of a major page such as this distracts from a tidy article overall. The size of this article was kept down by the fork. But I always expected that an appropriate link to the fork would be maintained, contrary to what happened until the link was replaced today.Rgdboer (talk) 23:06, 4 August 2009 (UTC)

## Etymology

I've added a bit from a source I found, though it could use some review and adjustment perhaps. Cohn notes Hilbert introduced the term in 1892, so that's good enough to quote that, but Cohn doesn't list any publication of Hilbert's but the 1897 publication of Die Theorie der algebraischen Zahlkörper in his bibliography. I don't read any German, so I can't really search for any sources or uses by Hilbert of the term before Zahlkörper being published. Also the relevant part of Cohn's book is viewable on Google books for anyone that doesn't have a copy. I don't have much background in modules, but I think I've distilled what Cohn wrote correctly for what is needed for this article. - Taxman Talk 15:08, 5 August 2009 (UTC)

## Simplify Introduction

I know a number of you have worked hard on this page, especially to include nontrivial details and lots of useful information. However, the introduction to this page is both intimidating and unusually long. I suggest most of the introduction information elsewhere, and leaving the intro to emphasize the following two facts:

1. Rings are a simple concept: a set with two operations, one like addition, the other like multiplication, perhaps without inverses.
2. Rings are ubiquitous: integers, real numbers, sets of polynomials, sets of functions, suitably defined, all form rings.

Your hard work may be ineffective if your page does not welcome the uninitiated. For us mathematicians, it's too easy to forget how bewildering our writing can sound to the layman. By writing an encyclopedia article, you have the chance to popularize the knowledge of specialists -- don't pass it up! Expz (talk) 17:22, 19 November 2009 (UTC)

I should note that the current lead is due to me. The point is that I wished to include some of the various applications of ring theory in other branches of mathematics, as well as ring theory in its own right. If you look through the history of this article, you will find various other versions of the lead, which address only the basic aspects of the theory.
Prompted by the above comments I have gone back to versions of the article which existed late in 2008. The lead there was much shorter and simpler, and I think this made the article less daunting for someone who did not already have knowledge of the subject. I am inclined to agree with Expz: the lead should be short and simple, and other information should come in later sections. JamesBWatson (talk) 11:51, 20 November 2009 (UTC)

Since we have all agreed that the lead should be shorter, I shall rewrite the lead tomorrow and shall request opinions. Please give me this time to do so. You may revise the new lead once it is finished (although you are, of course, entitled to edit the lead in its current state, I would appreciate it if you let me fix it first). Cheers, --PST 12:38, 20 November 2009 (UTC)

Do I detect a slight trace of WP:OWN?
In comparison to group (mathematics), the current lead seems to be of appropriate length. Opinions on the new lead? --PST 13:41, 20 November 2009 (UTC)
I like the draft much better! Very welcoming for the new-comer, yet still with useful leads to the one looking for deeper information.
Better there are people editing the article who've given it serious thought than everyone editing willy-nilly. Wikipedia always welcomes contributions, but what good to anyone are careless ones? Expz (talk) 18:42, 21 November 2009 (UTC)
Thanks! I do put a lot of thought into this article, but it always helps to have people point out some flaws, as you have done. Of course, the lead can still be improved, but mainly it is the second paragraph which needs improvement in my view. Unfortunately, I have not had much time for this article lately. --PST 00:24, 22 November 2009 (UTC)
Sorry, perhaps I wasn't clear the second paragraph was directed at the WP:OWN comment. Expz (talk) 02:43, 22 November 2009 (UTC)

## Recent changes

It was really about time that we had a few sections on polynomial rings and matrix rings, and thus it was great to see the recent improvements in that direction by an IP. I was however slightly skeptical about those changes because they did have a few technical errors; a result of a lack of formalizm. Nevertheless, I have expanded the article based on those IP's changes, by writing two new sections; one on polynomial rings and one on matrix rings. Of course, the subject of polynomial rings is too broad to go into depth in this article. Thus, one can only hope to give a formal definition followed by a few notes on its role in algebraic geometry, commutative algebra, and algebraic number theory. Likewise, the subject of matrix rings plays an important role in linear algebra and noncommutative ring theory, and so we can only hope to give a brief summary of its applications. Does anyone have any particular opinions as to what to include in these two sections? If not, I can proceed at some point in the future to expand these sections along the directions already mentioned. --PST 03:38, 8 December 2009 (UTC)

## Edit on 12:29, 31 March 2010

On 12:29, 31 March 2010 JamesBWatson reverted the edit of 07:12, 31 March 2010 which corrects following punctuation errors in the introduction of the article:

To qualify as a ring the set, together with its two operations, must satisfy certain conditions—namely, the set must be

1. an abelian group (also called commutative group) under addition; and 2. a monoid (a group without the invertibility property is a monoid) under multiplication;a[›]

3. such that multiplication distributes over addition.

"[R]ing the set" is a punctuation error, as it requires a comma after "ring". The M-dash after "conditions" should be a colon. The ordinal numbers followed by periods (e.g., "1.") in a running paragraph is a punctuation error: proper punctuation is to surround such list numbers with parentheses when they are within a running paragraph: e.g., "(1)". Ordinal numbers followed by periods is correct when the list is set off from the paragraph, like so:

To qualify as a ring, the set—together with its two operations—must satisfy certain conditions: namely, the set must be

1. an abelian group (also called commutative group) under addition, and
2. a monoid (a group without the invertibility property is a monoid) under multiplication,a[›]
3. such that multiplication distributes over addition.

In the article's source code, the ordinal numbers followed by periods are formatted as a set-off list with single line breaks, so in the code it looks like a set-off list, but it doesn't display as a set-off list. So perhaps whoever wrote this list in this manner was confused as to how it would be displayed and didn't bother to check the result of their edit.

The semicolon within "under multiplication; such that multiplication" is incorrect, as the "such that multiplication ..." is not an independent clause. Instead, a comma is required here in order for the current phrasing to be correct.

Due to these punctuation errors, the article will be edited to correct said mistakes.--71.0.146.150 (talk) 18:51, 31 March 2010 (UTC)

## Wrong statement in article?

In the section on homomorphisms, the article states that "If f is a ring homomorphism from (R, +, ·) to (S, ‡, *), the inverse image of the identity element 1S of ‡ (that is, all elements of R that are mapped to 1S by f) is a subring of (R, +, ·)." Is that really so? Take for example the homomorphism from Z to Z_4 (mentioned in the previous line of the article). The inverse image of the multiplicative identity of Z_4 is {...,-3,1,5,9,...}, which is not closed under addition, and thus cannot be a ring. Gabn1 (talk) 08:14, 26 May 2010 (UTC)

I agree. Perhaps this started as a claim about the inverse image of 0S which does hold in rings without 1 (which some people call rings), and was later "corrected" to the current form? I have marked the sentence as "dubious" for the moment, but since it's false it should obviously be removed rather soon. Unless someone has an idea what could replace it. Hans Adler 08:34, 26 May 2010 (UTC)
The statement in question defines 1S as the identity element of ‡, which is the addition in the second ring. This is a totally unhelpful notation, as 1S suggests a multiplicative identity. Thus where Gabn1 wrote "the inverse image of the multiplicative identity of Z_4", it was the inverse image of the additive identity which was required. In the general notion of "ring" (not required to have a 1) the statement in the article is perfectly correct, but expressed in a misleading way. If the statement is to stay then the notation should be changed from 1S to 0S. However, since the article mainly uses the more restricted concept of ring with a 1, the statement should be either removed or else rephrased to indicate it is using the broader concept of ring. On the whole I think it is a fairly unimportant detail anyway, and I shall remove it. JamesBWatson (talk) 10:07, 26 May 2010 (UTC)