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:
Thanks, Point-set topologist (talk) 12:05, 22 December 2008 (UTC)
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;
I tried doing that but it is still problematic. Could you please help out? Point-set topologist (talk) 13:13, 22 December 2008 (UTC)
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)
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)
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)
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)
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.
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.
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...)
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...).
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.
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.
7) "In this article, all rings are assumed to satisfy the axioms as given above." Not true: Lie rings are considered later on.
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.)
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.
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.
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.
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?
15) "The ring of all integers ((Z, +, ⋅)) consists of precisely one unit, namely 1." Wrong.
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.
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.
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...
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...
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.)
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.
22) "Furthermore, any finite integral domain is a field." Explanation/reference?
23) "Fields and integral domains are very important in modern algebra." Yes. It would be nice to hear something about them.
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.
Plclark (talk) 14:04, 22 December 2008 (UTC)
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)
Comments by Jakob.scholbach (talk) 13:10, 28 December 2008 (UTC)
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.
The prose, wording and layout is often not inviting to read.
Comments by RJHall: It has some good material but there are some issues and a few of the sections need improvement.
Thanks.—RJH (talk) 17:28, 8 January 2009 (UTC)
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)