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Ring (mathematics)

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A script has been used to generate a semi-automated review of the article for issues relating to grammar and house style; it can be found on the automated peer review page for December 2008.

This peer review discussion has been closed.
I've listed this article for peer review because I have put a lot of effort into improving it for the past 4 days or so (some 500 edits!). My main motivation was that ring (mathematics) is such an important concept in mathematics and should be as good as group (mathematics) (I compared the two articles and tried to get one to the standard of the other). I firmly believe that this article can become a good article once enough citations are added and some sections (particularly the section on the history of ring theory) is improved. I would greatly appreciate comments (and more importantly input) in the following areas:

First and foremost, the last section of the talk page of the article, gives some areas for improvements. I would appreciate if people could improve the article along those lines.

I would also appreciate it if people could add some citations to the article. Since I don't have access to a library in the near future, I would appreciate it if people could find some good texts on the subject and cite facts in the article using those texts. Citations are probably the most important at this point in time.

The history section needs to be completely re-written and well cited. I am not an expert on history so I would greatly appreciate it if people who are experts could improve this section.

A nice visual image at the top of the article would be greatly appreciated. The current image can then be moved to the bottom.

Any comments regarding how this article can get to GA would also be appreciated. I have read the policy regarding "Wikipedia is not a textbook" and have corrected the "we"'s (I am used to writing "we" but I will try not to on Wikipedia). I know that some sections may need contracting so if you could mention any such sections I would be grateful.

Thanks, Point-set topologist (talk) 12:05, 22 December 2008 (UTC)[reply]

Hi there. There have been a lot of edits, all on very few days! Here are the most obvious things I think can be improved;

Don't put main article template in every paragraph. I think the template is unnecessary if the concept is linked in the paragraph
I don't really like the "basic properties of rings" section. Perhaps make it look shorter by: 1) Writing the number of the theorem and the actual theorem on one line (e.g. Theorem 1: a0=0a=0) 2) Hiding the proofs in unwrap-able windows, as done there: [1]

I tried doing that but it is still problematic. Could you please help out? Point-set topologist (talk) 13:13, 22 December 2008 (UTC)[reply]

Open any book you have on ring theory and find references (author+page number)! It isn't that difficult (unless you have NO book, but that can't be because there is [2] and others).

The references section is huge! Only the citation section needs to be improved. I will use your book for citations (thanks!). Point-set topologist (talk) 13:13, 22 December 2008 (UTC)[reply]

I find the article a bit dry. You present examples and results but not much intuition is given.

I am aiming to give more intuition now (note that several examples such as the "integers" are given in the first section). I think that mainly intuition regarding the "concepts" must be given (like ideals, quotient rings etc...). Point-set topologist (talk) 13:13, 22 December 2008 (UTC)[reply]

Here is some historical information [3], if you're a bit dry on information.

Yes, I know some good historical sites but I don't think I have the time to write that much! If you could add some history yourself, I would greatly appreciate it (but I will add some history on my own later on when I have the time). Point-set topologist (talk) 13:13, 22 December 2008 (UTC)[reply]

I hope this is helpful. Randomblue (talk)

Thanks! I tried hiding the proof but somehow it does not come out correctly. Could you please help there? Point-set topologist (talk) 13:13, 22 December 2008 (UTC)[reply]

In my opinion the article is far from being GA class: most of it reads like Chapter 1 of a (somewhat dull) textbook on rings. The article does not convey the richness of the ring concept and its vital importance in mathematics. Except for the categorical description at the end (which, as a working algebraist, is not among the first 500 or so pieces of information I would want to convey about rings; I would be interested to know if other experts in the area feel differently), there is no information about 20th century results. Neither is there any hint that the commutative and noncommutative cases are vastly different, that the former has been successfully recast as a subfield of abstract algebraic geometry, and that the latter is related to geometry as well but in a way which is as yet much more mysterious.

On the other hand, here are some specific things that can be improved:

1) "such that these two operations are distributive over each other (i.e work together)." This is needless sloppy: first, "work together" does not have any precise meaning. Surely the vast majority of readers of this article will have encountered the distributive property in high school algebra. Moreover "distributive over each other" implies a symmetry between addition and multipication that is, of coure, incorrect.

Many people know distributivity, of course, but how many people interpret its meaning. I think that most people play around with distributivity without actually understanding why it is important. I corrected the other part (thanks!). PST

2) Although one still sees it in print occasionally, most contemporary expositors frown upon (and indeed, deride) the inclusion of "closure" as an axiom for a binary operation. Rather, this is part of the definition of a binary operation.

Actually, a similar thing is done with group (mathematics) which is FA. I am trying to model this article like that. PST

3) "Like with most other mathematical objects, there are often disputes as to what axioms a ring should satisfy." Too strongly worded; lots of mathematical objects have agreed upon definitions (e.g. group, partially ordered set, topological space...)

I totally agree; I changed this to "like with some axiomatic theories ..." (this also applies to physics). PST

4) "For instance, some authors insist that 1 ≠ 0 in a ring (in words, this means that the multiplicative identity of the ring must be different from its additive identity). In particular they don't consider the trivial ring to be a ring (see below)." Again, this is something that the majority of modern expositors consider to be a bad idea. For instance, given generators for an ideal I in a ring R, it is in general difficult to tell whether the ideal is the unit ideal, so if you adopt the above convenition you find yourself unsure of whether R/I is a ring or not. I would prefer if this statement were reworded as a warning to this falling-out-of-fashion usage, together with some citations as to where it is still used (not in any of the dozen or so algebra texts that I own...).

I didn't actually write that so I am unsure as to whether it is true. But it is just what some authors like to impose and we must include all possible definitions (but the main definition we follow in the article is the most common one). PST

5) "Rings that satisfy the ring axioms as given above but do not contain a multiplicative identity are called pseudo-rings." Please give a reference to a standard text where this terminology is used. It is not very familiar to me.

Me neither (it is not familiar to me either); I think this terminology was introduced by Bourbaki. Perhaps have a look at pseudo-ring. PST

6) "Note that one can always embed a non-unitary ring inside a unitary ring in the canonical manner (see this for a proof)." "the canonical manner" is ungrammatical. Moreover, it is not just a grammatical issue: there are several inequivalent ways of producing a ring with identity from a rng: the article on rng should give some references.

That is true. As I said, this was there before I edited the article, but I fixed it up now. PST

7) "In this article, all rings are assumed to satisfy the axioms as given above." Not true: Lie rings are considered later on.

...unless stated otherwise. PST

8) "An example of a non-commutative ring is the ring of n × n matrices over a field K, for n > 1..." It is not said what "R" is: presumably the real numbers is intended, but note that the construction works over any nontrivial ring. (It's not a good idea to assume without comment that R stands for the real numbers in an article on ring theory.)

Made explicit. PST

9) "Associativity of addition in Z4 follows from associativity of addition in the set of all integers." This is true, but it is not explained why. It would be good to discuss this in the context of quotient rings.

:Sometimes I hate explaining things that I know are correct but are difficult to explain without going outside the topic! But I will do this in the section on quotient rings and include a link (inside the article) to there. PST (I am bolding this to remind myself to do it).

10) "Somewhat surprising is that this does not hold for the ring (Z4, +, ⋅):" Encyclopedia articles don't tell you what is surprising.

Done. --PS T 17:31, 18 January 2009 (UTC)[reply]

11) "Note that since the non-zero elements in this ring form an Abelian group (the empty set is vacuously an Abelian group), this ring is a field and therefore an integral domain." NO!! This is a serious error. I will remove it as soon as I finish here.

12) "Simple consequences of the ring axioms" Is this section really necessary in the main article on rings? Remember, this is an encyclopedia article, not chapter 1 in a textbook.

Well, it is interesting that the product of 0 with any element of a ring is 0; or that the additive inverse of a equals a times the additive inverse of 1. Proving these properties effectively requires you to use distributivity which is a key axiom in ring theory. It think that these facts should be mentioned. Since the proofs don't clutter the article, I don't see the harm (some people are against proofs in WP but personally I think the opposite if the proofs are crucial to the article) in having them. --PS T 17:31, 18 January 2009 (UTC)[reply]

13) "An integer in a ring is..." this is decidedly nonstandard terminology, which conflicts with the use of "integral elements" in ring theory. At the very least a citation is necessary, but best would be to not use the terminology altogether.

I didn't write that. Bolding this statement to remind myself to do something about it. --PS T 17:31, 18 January 2009 (UTC)[reply]

14) "Binomial theorem for rings" Again, is this one of the most important things to include in the survey article on rings? If so, why? Where is the context?

Agreed. As usual, I am bolding this statement... --PS T 17:31, 18 January 2009 (UTC)[reply]

15) "The ring of all integers ((Z, +, ⋅)) consists of precisely one unit, namely 1." Wrong.

Oops! I forgot -1. --PS T 17:31, 18 January 2009 (UTC)[reply]

16) "Ring morphism" Notice that the article that it links to is in fact "ring homomorphism." That is more standard terminology, especially for an elementary/generalist approach.

I noticed that you changed that. Thanks! But I noticed also that you defined a ring isomorphism by saying that f has to be a morphism as well as invertible but you also referred to category theory. Note that it would better to use the notions of "epimorphism" and "monomorphism" because in general, you cannot define isomorphisms in this way unless you are dealing with a concrete category. --PS T 17:31, 18 January 2009 (UTC)[reply]

17) "An isomorphism of rings is a bijective ring morphism." This makes the opposite mistake (although, to be sure, a very common one.) Once you have defined a category -- objects and morphisms -- you don't need to give an independent definition of an isomorphism: it is always a morphism which has a two-sided inverse. With this definition, the fact that a homomorphism of rings is an isomorphism iff it is bijective is a (simple) theorem about rings, not a definition. Defining things this way makes it harder for the reader/student to adjust to (concrete) categories in which bijective morphisms need not be isomorphisms, like topological spaces.

Oooops! I read 16) before 17) so you might as well ignore my previous post. --PS T 17:31, 18 January 2009 (UTC)[reply]

18) "The product is natural because the quotient ring R X S / R is isomorphic to S and similarly R X S / S is isomorphic to R." Huh? Again, here is a place where category theory might actually be helpful: the direct product is natural because it satisfies a certain universal mapping property...

I am a bit worried about complicating matters. But feel free to add what you know and we can position it in the right manner. --PS T 17:31, 18 January 2009 (UTC)[reply]

19) "There is a one-to-one correspondence between Boolean algebras and Boolean rings and hence the name 'Boolean'." Reference?? Why was the concept of a Boolean ring introduced? What role does it play in the the larger picture of ring theory? No context is provided...

I would think that this result is trivial... :). But I guess I should have given a reference. I guess I have to remove a few sections later (bolding this to remind me). --PS T 17:31, 18 January 2009 (UTC)[reply]

20) "The endomorphism ring of an Abelian group is trivial if and only if the Abelian group in question is the trivial group." Explanation/reference? (It's easy: there is always the identity endomorphism and the zero endomorphism.)

Trivial as you noted. --PS T 17:31, 18 January 2009 (UTC)[reply]

21) "Rings with additional structure" This section is very poorly organized, in that the three examples are of three different sorts: (i) a Lie ring is not a ring according to the conventions set up in the article, so it does not have additional structure, it has different structure. (ii) A topological ring is indeed a ring with additional structure; (iii) integral domains and fields are not rings with additional structure; they are classes of rings with additional properties: e.g., they live in the category of rings, unlike Lie rings or topological rings (at least, before one applies a forgetful functor). One would think that integral domains and fields should be given more prominence in the article and should be discussed earlier in and more depth.

Yes, I must do something about it. --PS T 17:31, 18 January 2009 (UTC)[reply]

22) "Furthermore, any finite integral domain is a field." Explanation/reference?

Trivial. But I will explain it. --PS T 17:31, 18 January 2009 (UTC)[reply]

23) "Fields and integral domains are very important in modern algebra." Yes. It would be nice to hear something about them.

What particularly did you have in mind? (note that there is so much I could say about them that it would no longer be an article about rings! I know some important concepts (we could talk about the fundamental theorem of algebra (e.g Galois theory) or polynomials or algebraic geometry or algebraic number theory etc...)). --PS T 17:31, 18 January 2009 (UTC)[reply]

24) There are too many references given. The difference between a reference and further reading is that a reference should actually be referred to somewhere in the article. It does not seem that that is the intention here. Moreover, the list looks rather haphazard.

It is alphabetically ordered but at least there are not too few references! --PS T 17:31, 18 January 2009 (UTC)[reply]

Plclark (talk) 14:04, 22 December 2008 (UTC)[reply]

I will respond to all your comments later but just one note: Before I edited the article, there was a lot of (incorrect) content. Some of it, I have re-written but I have not yet re-written all of it. I will go through the things I copied (from the old version) and check for errors now (there seems to be a problem with the article in this respect as you have mentioned). Point-set topologist (talk) 14:51, 22 December 2008 (UTC)[reply]

Comments by Jakob.scholbach (talk) 13:10, 28 December 2008 (UTC)[reply]

I have done an informal review earlier (at the article talk page), so here are some further comments. I also concur with Plclarks comments above.

As an example, I did one edit to the section on the center of a ring. Most others require at least this type of improvements.

Thanks! I will look to do this to each section. --PS T 07:39, 19 February 2009 (UTC)[reply]

- It is often reasonable to start a section or subsection with a motivational statement what the notion is about. (I added " There are various notions to adress the non-commutativity of general rings R.")

Thanks! I will do this to each section. --PS T 07:39, 19 February 2009 (UTC)[reply]

- Repeating: "Let (R, +, *) be a ring" every time is superfluous.

It will take some effort to undo this but we should certainly aim to in the later stages of the article. Bolding this to remind me. --PS T 07:39, 19 February 2009 (UTC)[reply]

- Also, I find it a bit awkward to have "we", "one" all over the place (this also contradicts MOS).

I will start a long run to remove these. Bolding this to remind me. --PS T 07:39, 19 February 2009 (UTC)[reply]

- Trivial statements should be avoided. (I removed " If R is a commutative ring then R equals its center; that is, if C is the center of R then C = R. Therefore, any subring of a commutative ring, R, is central in R."). This is simply to make sure that when adding more involved content, you don't get space problems.

OK. --PS T 07:39, 19 February 2009 (UTC)[reply]

- "Intuitive" statements should be done with great care: "Intutively, the center of a ring can be thought of the 'maximal commutative subring' of that ring" conveys the wrong intuition: a ring may have commutative subrings but the center may be trivial.

I am not sure what you mean by "the center may be trivial". The center of a ring always contains the cyclic subgroup generated by 1 (under addition) so the only time this can be trivial is if the ring itself is trivial. --PS T 07:39, 19 February 2009 (UTC)[reply]

The prose, wording and layout is often not inviting to read.

"Integral domains and fields" does not belong to "Rings with additional structure" IMO, rather it is a special case.
I really miss a big section titled "Noncommutative rings" and one "Commutative rings". ("Integral domains and fields" should then be put in the latter). In general, the article is currently really biased (a no-go in WP) towards the commutative case.
I think the definition section is highly repetitive. This is in line with a general criticism also uttered by the above review and other discussions: the audience of the article is currently too much slanted towards somebody who has never heard about anything in algebra. Some parts of the articles should be taylored for such an audience, but the article misses any content that is inviting or interesting for more advanced readers. This means that the article is not "broad" (a GA criterion). Also, it must be said that big parts of it repeat glossary of ring theory. Jakob.scholbach (talk) 13:10, 28 December 2008 (UTC)[reply]

Comments by RJHall: It has some good material but there are some issues and a few of the sections need improvement.

I think the illustration at the start needs a clearer explanation.
Citations should follow punctuation without a gap. See tags [1], [2], [3], [4], ...
Is the wording "Although in some naive sense ... subject to certain equivalences)," needed? I think if you begin the sentence with the component "In many respects the...", it is sufficient without being offensive to the reader.
Per WP:MTAA: in the "Formal definition" section, the notation "+ : R × R → R" needs to be explained somehow (such as with a wikilink to Function_(mathematics)#Notation).
Check for spelling errors, such as "identitiy".
For parts of the "Notes on the definition" section, see User:Tony1/How_to_satisfy_Criterion_1a#Eliminating_redundancy. For example, "disputes" works just as well as "often disputes". For "note that", see WP:EDITORIAL. &c.
The "History" section needs improvement. There are too many brief paragraphs. For example, "Richard Dedekind (image to the right) introduced the concept of a ring" should give a year and possibly describe why he introduced the concept.
The inline links should be converted to proper citations.
The "Basic concepts" section has too many subdivisions and bullets.
Can the template in the "Category theoretical description" section be addressed?

Thanks.—RJH (talk) 17:28, 8 January 2009 (UTC)[reply]

Comments by Taxman: I commend you on working on such an important and yet difficult to do well article. Indeed you have picked a good model by looking at Group (mathematics), though of course even that can be improved. You have a lot of great advice above, so I thought I would just make a comment on the coverage of the article. In order to be a clean overview of the topic, some material must be cut and moved out to more detailed sub articles as per WP:SS. There's too much there right now to allow space for the most important things to be developed properly. Of course it's hard to prioritize, but my suggestion is to look at a several different textbooks at different levels (undergrad, grad, etc) and look at what they collectively consider most important. All the rest necessarily must be covered in other articles. The good news is once this is done some of your work is easier because you're not working on the wrong things. - Taxman ^Talk 14:51, 17 January 2009 (UTC)[reply]