Talk:Ring (mathematics)

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Mathematics grading: B Class High Importance  Field: Algebra
Examples, history, motivation. Tompw 13:13, 5 October 2006 (UTC)

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[edit] Associativity and unit?

From my experience, the definition of a ring that does not include associativity and the existance of a unit is the most common. Wouldn't it be advisable the encyclopedia be changed to match that definition, or are there objections to this? --- schnee (20:51, May 24, 2003 (UTC))

I prefer the more general definition, but I'm not sure it's the most common. Changing the definition used by Wikipedia would take quite a lot of work, as there are a great many articles which mention rings and almost all of them would need to be reworded. Even listing all articles that need to be changed (which is the necessary first step) would be a fair amount of work. In any case, there would first need to be a consensus that this is the right thing to do. --Zundark 21:08 24 May 2003 (UTC)
I agree, it would certainly be a lot of work. Who would have to be asked for a concensus on this change, though? Also, on an unrelated note, is it actually being made sure that the definitions used in the articles match the one given on this page? --- schnee (22:30, May 24, 2003 (UTC))
You would need to get agreement from most of those who are involved in editing the mathematics articles, particularly Axel Boldt, who has generally been the active. For myself, I feel sure that it's not usual to omit associativity, so I would be against making the change that you suggest.
As for your other question, yes we do try to make sure that Wikipedia uses consistent terminology in mathematics articles. --Zundark 10:47 25 May 2003 (UTC)
Isn't most of the interesting stuff on rings (that we'll cover) about the associative unity rings? We've mentioned the alternate usage, and any articles that wish to speak of non-associate rings should probably specify that anyway -- most of the books I've seen define rings as associate and with unit. -- Tarquin 22:40 24 May 2003 (UTC)
Most important rings are associative, but there are some exceptions (Lie algebras, Jordan algebras, the octonions). On the other hand, any book on ring theory has to cover ideals, which are non-unitary rings. --Zundark 10:47 25 May 2003 (UTC)

Personally, I'd prefer leaving the definitions as they are: for non-associative non-unitary thingies we could make algebra over a commutative ring (a module with a bilinear operation); most examples already fit under algebra over a field.

There's another issue: if we did change the definition to encompass non-unitary rings, we'd have to change the definition of "ring homomorphism" and would have to check all uses of that term to see which ones need to be changed to "homomorphism of unitary rings", since the two concepts are different. AxelBoldt 01:04 26 May 2003 (UTC)

[edit] Possible Definition Contradiction

I have an observation that may be the result of my limited knowledge of abstract algebra: the definition of rings notes that the commutative law is not an axiom of rings, but the definition states also that a ring is an abelian group. The definition of an abelian group states it is a group that is commutative. These two definitions appear to contradict each other. Can someone add some clarification? : Clif 20:05, 26 Nov 2003 (UTC)

It says that commutativity of * is not an axiom. (R,+) is abelian, so + is commutative. There is no contradiction here; these are two different operations. --Zundark 08:55, 27 Nov 2003 (UTC)


[edit] Definition of ring and unit elements

When I was a graduate student in pure mathematics , the common definition of a ring did not include the presense of a unity element or the requirement of the mapping of unity element onto corresponding unity element by a homomorphism, although I believe this is the accepted definition by the Bourbaki school. My research was in certain areas of ring theory S. A. G. (comment by anon IP 152.163.252.197 19:03, May 8, 2004 (UTC))

Rings without identity aren't too uncommon. Moreover, speaking to some people who do research in what they call "ring theory", it doesn't appear this is an assumption. Of course, for people like myself, in number theory, "ring" almost always means "commutative with identity". But on these types of articles, can't this just be said once at the beginning of the article? "Non-unital ring" is somewhat deceptive, because non-unital is not required, "non-unital" just means "admitting the possibility unity doesn't exist", not "unity doesn't exist". Or at least, that is my understanding. Associativity seems more standard...they are examples of non-associative rings, esp. in Lie theory, but these seem less widespread than non-commutative. Revolver 02:01, 11 Jun 2004 (UTC)
Comment: I was recently told at another article that encyclopedia-wide conventions aren't encouraged, i.e. Wikipedia shouldn't have a "universal convention that a ring means such and such", in this case, this article should simply say "In this article, rings are assumed to...", NOT "In Wikipedia, we assume..."Revolver 02:03, 11 Jun 2004 (UTC)
I see I misread, associativity is assumed, but no commutativity. Still, I'm not sure why having identity is so important, isn't nZ = { nz : z in Z } usually considered to be a ring?? Revolver 02:18, 11 Jun 2004 (UTC)
It depends on time and place. The University of Cambridge changed to require a 1 in about 1984. --Henrygb 23:07, 30 Apr 2005 (UTC)

[edit] Striving for conventions

I'm coming to doubt whether it's a good idea to strive for universal math defintiions on wikipedia. Say, you define a ring to have unity. Then, what about if there comes around a flurry of articles on non-unital ring theory (quite possible, it's a big research area). Either all these articles have to violate the convention, which is not a good idea, or they have to always say, "non-unital ring" 500,000 times, instead of saying "in this article, 'ring' does not require a unity". The latter seems to make much more sense to me. Revolver 02:26, 11 Jun 2004 (UTC)

I agree that an article should be able to override the Wikipedia definition if it needs to, so there's no need to say "non-unital ring" 500,000 times. But we really do need to have a Wikipedia definition, because otherwise every article that mentions rings would need to say whether or not its rings are assumed associative, and whether or not its rings are assumed unital. This would be a major pain. --Zundark 11:17, 11 Jun 2004 (UTC)

[edit] small suggestion

If you decide to stick with the current definition you might want to add 2Z as a non-example and mention that it is an ideal but not a subring in Z ~reader

[edit] References for definitions

General comment about using references for definitions : In general it would be a good idea to use a reference source and list it for any definition used in pure mathematics articles. S. A. G. (comment by User:Remag12@yahoo.com 15:19, Aug 15, 2004 (UTC))

[edit] Article is FULL of errors

This article is FULL of errors! A ring first of all is NOT an abelian group. It is not at all required to be commutative, and nor does it have to have an identity (unit) in this case, nor does it generally have inverses. To settle the discussion further down (I moved this comment form the top of this page to the bottom, so "further down" should be read as "above" — Paul August ☎ 13:55, May 26, 2005 (UTC)), rings are however generally considered associative, as this is a requirement for a binary operation, and a ring is based around two binary operations. (Unsigned comment by anon IP 212.169.96.218 13:17, Mar 26, 2005 (UTC).)

Associativity is NOT a requirement for a binary operation. Subtraction is a binary operation, and it is not associative. Nahaj 13:38, 1 October 2005 (UTC)

The article does NOT say the a ring is an abelian group. It says that (R,+) is an abelian group. There's a big difference between the two. (Unsigned comment by 68.108.171.192, 18:50, September 29, 2005 (UTC).)

[edit] Definition: some references, remove unit element?

I've just pulled off my library shelves the three books I learned algebra from, when I was a grad student, many moons ago. They are:

  • Clark, Allan, Elements of Abstract Algebra (1971)
  • Goldhaber, Jacob K. and Ehrlich, Gertrude, Algebra (1970)
  • Herstein, I. N., Topics in Algebra (1964)

In regards to multiplicative associativity:

  • Clark, and Goldhaber and Ehrlich defines multiplication to be associative. Clark saying this is "customary"
  • Herstein: Defines what he calls an associative ring, with the remark that "whenever we speak of ring it will be understood we mean associative ring. Nonassociative rings … do occur in mathematics and are studied, but we will have no occasion to consider them"

In regards to a multiplicative identity element:

  • Clark defines a ring as having a "unit element", however he goes on to remark that this is not "customary", but that all the rings he will talk about will be rings "with unity" and it will be convenient not to have to always say "ring with unity"
  • Goldhaber and Ehrlich, and Herstein do not require rings to have identity elements calling such, a "ring with identity" or "ring with unit element", respectively.

In regards to multiplicative commutativity:

  • Clark, defines multiplication as commutative, however (as with identity elements above) says that this is not "customary", but is done for convenience sake.
  • Neither G&E or Herstein define rings to be commutative.


So we have the following:

Author associativity identity element commutativity
Clark yes yes but yes but
Goldhaber and Ehrlich yes no no
Herstein yes but no no

Based on the above I think we should remove the requirement for a multiplicative identity. Anybody have other (better?) references to cite?

( I left the above comment on 15:34, May 26, 2005, but I forgot to sign and timestamp. Paul August ☎ 16:09, May 26, 2005 (UTC))

Please keep the unit element. It's not just the ring theorists to consider. I'm sure you'll find that Bourbaki includes a unit element. About the only reason not to, is to be able to talk about any ideal as a subring. This is not a big advantage. Charles Matthews 15:50, 26 May 2005 (UTC)
It is my impression that in number theory and algebraic geometry rings are always assumed to have a unit. The main proponents of unit-less rings seem to be ring theorists, but they are not a majority, and we do not have to follow their conventions. I prefer unital rings as the default, and articles can have the choice of specifically overriding this default by mentioning their nonstandard terminology at the beginning of the article. I think most "nonassociative rings" can be treated as nonassociative algebras over rings instead. - Gauge 00:27, 3 October 2005 (UTC)
As I said above, conventional definitions change: in Cambridge around 1984, i.e. after your books were published. --Henrygb 10:59, 12 October 2005 (UTC)


[edit] Some examples are not really examples

Some examples seem to be really properties/theorems. For instance:

  • A ring (in the categorical sense) is commutative iff it is equal to its opposite ring.

This is not really an example. Maybe this one (and other like this) should be moved to a special 'Theorems' section.

Moreover, I think that the article should keep to a few simple and illuminating examples, that maybe should have some comments to attract the non-mathematician. I could give it a try, but someone else's opinion would help

I'm waiting for another opinion before going ahead and changing it. - AdamSmithee 21:24, 22 November 2005 (UTC)

The "example" you quote above is not only not an example, it doesn't make sense. I suggest deletion. Rick Norwood 21:43, 22 November 2005 (UTC)

Unfortunatelly I know nothing about categories, so I can't say whether it makes sense, but I also think that it shoud be deleted (see below).
Anyway, I think that there are two types of problems with the examples:
1. some of them are not examples but properties/theorems (as I already said)
2. some of them are too advanced. The utility of examples is to help people who know nothing about rings get a sense of what we are talking about. So examples (in my opinion) should be simple and they should be taken from areas about whitch the user is most likely to already know something. Someone who knows what a Weyl algebra is (for instance) would probably not need ring examples. These advanced examples serve no use and, worse, are just plain scary
These are the examples that I suggest to be deleted or moved
a. endomorphisms of a group - not example
b. endomorphisms of abelian category - not example + advanced
c. holomorphic functions - advanced (do we need examples from complex analysis?)
d. group ring - not example
e. free algebra - advanced - this is a complicated way of giving an example of a noncomutative ring; we already have matrices, which is a straightforward example, and we could use some simple ring of functions with function composition if we really want another example of noncomutativity
f. Weyl algebra - advanced
g. tessarines - advanced
h. path algebra of a quiver - advanced
i. ring in the categorical sense - advanced + possibly incorrect?
I would like to have another opinion on each one of these, as I hesitate too delete someone else's work, but I think it would make the article better - AdamSmithee 09:50, 24 November 2005 (UTC)

My vote is to keep a and d, delete the rest. Rick Norwood 13:25, 24 November 2005 (UTC)

Keep a.--Patrick 14:38, 24 November 2005 (UTC)

Rick Norwood and Patrick, thank you both for the feedback. I'm very new to Wikipedia and I really needed second opinions. I'll go ahead with the modifications now. To stay on the safe side and not delete potentially useful content, I'll keep a and d as each of them got at least one 'keep' vote. The rest will have to go. - AdamSmithee 21:46, 24 November 2005 (UTC)

While I agree with the deletion, as a general rule it is a good idea to wait 24 hours when you ask for a response. Different people are on line at different times during the day. On the other hand, you are moving with admirable caution -- a lot of new users just delete right an left without even asking...and the deletions get reverted. Rick Norwood 22:46, 24 November 2005 (UTC)

I don't think we should delete correct examples just because they are "advanced". First of all "advanced" is a relative term. One persons "advanced" is another persons "basic". Second, these aticles are for a wide range of audiences, "advanced" as well as "beginner". Paul August ☎ 00:31, 25 November 2005 (UTC)

You are right, I should have waited 24 hours. I just didn't realize that there wasn't enough time for people to answer... But, if it turns out that there are good reasons to keep some of this examples, I will put them back.
Paul August, generally I agree that one person's "advanced" might be another person's "basic". The point is that some examples are 'advanced' relative to the concept of a ring - so, if the concepts in these examples are 'basic' for someone, chances are that rings would be 'basic' for that person too, so she would not need examples. The examples I noted are generally from the same area as rings, abstract algebra, and I think that we can agree that in abstract algebra rings are a 'basic' concept. Realistically, it is highly unlikely for someone to know advanced abstract algebra while having never heard about rings. On the other hand, (advanced) examples from other fields, especially fields outside math (like phisics maybe?) marked as such might be usefull indeed. - AdamSmithee 15:48, 26 November 2005 (UTC)

[edit] The new definition of a subring.

The new definition of a subring, like the old definition, is false. In the ring of integers, the set of natural numbers is closed under addition and multiplication, and contains the identity element, but is not a subring, because it does not contain additive inverses. The correct definition is closed under subtraction and multiplication. Rick Norwood 02:04, 9 December 2005 (UTC)

Woops. Thanks. I guess it might be better to say that it must form a ring under the same operations and identity. -- Fropuff 04:03, 9 December 2005 (UTC)

[edit] A solid reason (I think) to reject requiring unity

Suppose that however otherwise you define a ring, you require that it have unity. Then by that definition, the integers can have no non-trivial subrings, since any subring containing 1 also contains the (additive) subgroup generated by 1, namely all of the integers. But I have a hard time beleiving any reputable source on ring theory wouldn't consider the integers to have lots and lots of subrings (the evens, the divisibles by three, and so on).

You could try and remedy this by calling these "subrings" something else, but that seems unduly complicated and confusing. They look like rings to me. Further, this problem generalizes to every ring with unity, not just the integers, in the following way. Central to the study of a ring are its 'ideals'. For those rusty on the subject, ideals are subrings that are roughly analogous to the group theory concept of normal subgroups. Ideals are precisely those subrings which appear as kernels of homomorphisms, and you can mod out a ring by an ideal to obtain a new ring. Now, here is the problem: if you define a ring to have unity, than NO ring can have any non-trivial proper ideals (it can be shown quickly that an ideal containing 1, or indeed any unit, must be the entire ring). Again, you can define an ideal as something other than a subring. But its unduly complicated and counterintuitive.

Though I don't know much about category theory, the problem seems to go to the heart of the categorical foundations of ring theory. Any category of objects has morphisms between the objects (homomorphisms between rings). To have the kernel of a homomorphism of rings NOT be a ring itself I think would introduce some really serious problems, though I can't be more specific than that.

One gains nothing by strictly defining a ring to have unity, other than not having to say "ring with unity" all the time. No doubt what the authors previously quoted meant is that they will always be starting with a ring with 1, but keeping in mind that the subrings of a particular (unital) ring might not necessarily contain 1 (the interesting ones usually won't).

So I think that the unital condition should be completely dropped. I think commutativity of multiplication should also be dropped -- the ring of n x n matrices over a ring is just too important and natural an example to lose. I think associativity of multiplication should be kept however, since I think it would be in general difficult to derive any useful and meaningful theorems without associativity. As someone brought up, non-associative rings are an active and important field of study, but even then one usually replaces associativity with some other condition (e.g., the Jacoby identity in Lie Algebras). To just throw it out doesn't leave you with much to go on. Michael C 08:30, 27 March 2006 (UTC)

This subject keeps coming up, but since Bourbaki defines rings without the requirement of an identity and Herstein defines rings with the requirement of an identity, the best we can do is say that major authorities differ. Not to mention both would be original research. The evens, multiples of 3, etc are a "something" and that something is a subgroup. Using the Bourbaki definition, they are also subrings. I don't know of any authority who requires ring multiplication to be commutative, but most authorities require ring multiplication to be associative. "Non-associative rings" are not rings, just as simigroups are not groups. Rick Norwood 14:04, 24 March 2006 (UTC)
I happen to have a copy of Herstein's "Topics in Algebra". His definition does NOT include the requirement of unity, as explicitly stated on pg. 84 (perhaps you're referring to a different book of his). This only goes to illustrate my point: that when authors "require" rings to have unity, they are not really doing so. They are merely freeing themselves of the responsibility of saying "Let R be a ring with unity" when beginning a theorem or example with an arbitrary ring. This is an understandable convention, especially in an introductory book on ring theory, where a whole host of powerful theorems requiring many pages can be developed for rings with unity, ones that do not necessarily hold for rings without unity. A similar situation might occur in a book on group theory: a particular section might be completely devoted to abelian groups, and they would say something like "all groups will henceforth be assumed to be abelian". But no one actually requires the definition of a group to include commutativity. (of course, in this case, one could conceivably define a group to be commutative, and still be internally consistent, since all subgroups of an abelian group are themselves abelian. The problem we're dealing with is much more serious, namely that certain subsets of rings which, as far as I can tell, NEED to be considered to be rings cannot be rings when we require them to have unity.)
Perhaps I have simply not read enough books on the subject, but I have yet to find an author who would not consider the evens to be a subring (not just a subgroup) of Z. In all the sources cited, I would suggest that the sources be checked again to see if "requiring" rings to have unity isn't instead the situation I just described. And not just for the purposes of this article, I would actually like to personally know how they work their way around the difficulties I described in the preceeding post, since I really can't imagine how they would do it. If I'm wrong, it wouldn't be the first time, but in this case I just can't see how. Michael C 08:30, 27 March 2006 (UTC)
With regard to Herstein and Bourbaki, I think Rick has inadvertently reversed the situation. As 72.240.171.117 says, Herstein's Topics in Algebra, does 'not require rings to have a multiplicative identity (p. 83), instead calling a ring that does, a "ring with unit element" (p. 84). I don't have handy access to Bourbaki, but our article says that they do require it. While we definitely need to state (as we do) that some authors require a unit and others don't, we have gone farther, choosing to define rings as Bourbaki, rather than Herstein. One could reasonably argue about whether that is the best choice for the purposes of our encyclopedia. (Myself, having grown up on Herstein, I think of rings the way he taught me.) Paul August ☎ 17:15, 24 March 2006 (UTC)
Sorry to keep harping on this, but I need to restate my position once more for absolute clarity. The argument for requiring rings to have unity seems to be that some authors do it. My contention (or suspicion rather) is that NO author actually requires a ring to have unity. Rather, that when seeming to do so he is merely saying that he will be restricting himself to examples of rings having unity. But I still think that such an author would allow certain subsets of rings with unity (e.g. the evens inside Z) to be considered rings. One could easily decipher the true intention of the author by looking at his discussion of the integers, to see whether he considers them to contain any non-trivial proper subrings. Again, I could very well be wrong, and this could all reflect a lack of imagination on my part as to how to deal with the problem. But its a possibility I think worth looking into before the debate proceeds further, since we might simply be misinterpreting the intention of such authors, making this whole debate moot. Michael C 08:30, 27 March 2006 (UTC)
Don't worry about the harping, I think this is an important issue to get straight. For my own part I will just say, that I think I understand your position and I don't disagree with it. I can add that all of my algebra texts, those given above as well as McCoy's Rings and Ideals (1948) and Herstein's Noncommutative Rings (1968) seem to support your position. However I don't know any more beyond what I've already said. I think we need some algebraists to weigh in. Paul August ☎ 18:19, 24 March 2006 (UTC)

Thanks, Paul, for understanding what I intended to say, instead of what I said. Yes, it is Bourbaki that requires an identity, and as far as I know he is the only author who requires a ring to have an identity, but he is influential enough, especially in Continintal Europe, that he cannot be ignored. Pretty good for a mathematician who doesn't even exist. Rick Norwood 21:17, 24 March 2006 (UTC)

If you don't require an identity in commutative algebra, the whole subject goes wrong. That is, you can't make prime ideals precisely those giving quotient an integral domain, and so on. This is fundamental stuff. As for ring theory in general, much less can be said, so you can perhaps get by without requiring a 1. But that's more a tribute to the difficulty of non-commutative algebra than anything else. Charles Matthews 21:23, 24 March 2006 (UTC)

Certainly, Bourbaki is not the only author to require an identity. See:

  • Cohn (2002), An Introduction to Ring Theory
  • Rowen (1991), Ring Theory
  • Lam (2001), A First Course in Noncommutative Rings

These are some random examples I found browsing Amazon. As to the "solid reason" given above, having ideals not be subrings seems to be a very minor annoyance compared to the gains one obtains by requiring an identity. -- Fropuff 22:19, 24 March 2006 (UTC)

Although Herstein does not define a ring to require unity, everything he does afterward does require it. My book is on campus, but I believe he says something to the effect that "from here onward it will be assumed so." Just my two cents. grubber 02:21, 26 March 2006 (UTC)

It seems I was premature in my assessment that authors requiring unity are only doing so for pedagogical and semantic convenience. I just came across a couple papers online, and indeed one such makes it clear that kernels of homomorphisms are NOT in general subrings (because they may not contain 1). This seems highly disturbing to me, but then again so does the concept of women voters. Welcome to planet mars.

I'd like to hear more from the opposing side, namely what one gains by requiring unity. Charles Matthews alluded to "the whole subject of [communative] algebra going wrong". I think I've more than elucidated my concerns as to what one loses, so I'd like understand a bit more of why other authors feel so compelled.

Finally, one last comment. I don't know which side this tends to support, if either, but I learned recently that any (non-unital) ring can be embedded in a ring with unity. If R is a ring, and Z the integers, the new ring is the set S = R x Z. S has the usual additive structure of R x Z, but with a somewhat screwy multiplication, in which the element (0,1) acts as multiplicative identity. Just something to think about. Michael C 08:46, 27 March 2006 (UTC)

This is fun to talk about, but it is, after all, a matter of definition, which is arbitrary, and which neither we, nor anyone else, has the power to settle. Rick Norwood 19:56, 24 May 2006 (UTC)

Well, you lose the idea that the integers are the initial object of the category of rings. Charles Matthews 22:16, 24 May 2006 (UTC)

If I'm doing commutative algebra (and by extension, algebraic geometry or number theory), I pretty much always assume that rings have 1. Like Charles says, otherwise stuff breaks. In some other areas I suppose it's not necessary to have a 1, but it's not something I do with any regularity. One interesting issue to think about is what happens with subrings... for example take the direct sum R = Z \oplus Z of Z with itself. Each copy of Z is an ideal in R, and is a ring in its own right; but the identity of each copy of Z is not the identity of R itself (although it is an idempotent). If you insist that rings must have 1, then presumably you also insist that subrings must share the same 1, so the two copies of Z won't be subrings of R; but if you don't ask for rings to have a 1, then you don't care whether a subring has the same 1, so the two copies of Z will be subrings of R. So this is a slightly more confusing situation than with 2Z living in Z. Dmharvey 02:57, 25 May 2006 (UTC)

[edit] Name

Why is a ring called a ring? Is there anything vaguely ring-like about a ring? —Ben FrantzDale 06:23, 22 September 2006 (UTC)

The name ring goes back to the example of the integers modulo n. In the integers modulo 6, for example, the integer 5 is followed by the integer 0, so the snake swallows its tail, forming a ring. (This is folk etymology. I have no information on the history of the word.) Rick Norwood 13:29, 5 October 2006 (UTC)
From wiktionary:ring: The mathematics sense was introduced by mathematician David Hilbert in 1892, a contraction of the German Zahlring. (Reference: Harvey Cohn, Advanced Number Theory, page 49.)
From German Wikipedia: the name "ring" does not refer to something that has a circular form, but rather to the union of elements in a whole. This meaning has been largely lost in the German language; some expressions, however, still retain the old meaning (e.g. "Deutscher Ring", "Weißer Ring", or expressions such as "Verbrecherring").
-- 131.111.8.104 03:10, 8 October 2006 (UTC)

[edit] Multiply by zero

How exactly does one prove that 0*a=a*0=0? Something along the lines of a*(b+-b)=b...? --67.10.175.242 22:15, 7 December 2006 (UTC)

0 = 0 + 0 so a0 = a(0 + 0) = a0 + a0. Subtracting a0 from both sides of a0 = a0 + a0 gives 0 = a0. -- Fropuff 04:18, 8 December 2006 (UTC)
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