Talk:Ring (mathematics)/Archive 1

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)

I believe the "rings" and "algebras" are being slightly confused here. The octonions--for example--are not a ring, but rather a normed division algebra. On the subject of the identity, as we've said, rings are not required to have them, and--though it will be effort-consuming--it is only right that Wikipedia reflect this. I agree that unitary ring homomorphisms are different, however I also believe it is unnecessary to make the distinction, as one can derive the necessarily true condition that a ring homomorphism between unitary rings preserves the unit. Do we know which pages the correction of the definition will materially affect (i.e. which pages will need to be appropriately altered)? Guardian of Light (talk) 17:46, 10 May 2009 (UTC)
 * It is not the case than a (not necessarily unital) homomorphism f between unital rings R and S preserves the unit. All one can conclude is that the image of the unit of R is a unit of f(R), not that it is the unit of S. Consider for example the map Z → Z×Z mapping n to (n,0). Algebraist 17:53, 10 May 2009 (UTC)


 * I am not sure if you are aware that you have just reacted to a discussion that is almost exactly 6 years old. --Hans Adler (talk) 18:16, 10 May 2009 (UTC)


 * No, but that's quite funny. I was examining the ring page and found an off definition of the ring, but wanted to check the talk page before editing it.  Didn't think to check the date.  ^_^ Guardian of Light (talk) 19:50, 11 May 2009 (UTC)


 * We were not (all those years ago) confusing rings and algebras. The point was that some definitions of rings require neither associativity nor a multiplicative identity. With such a definition, the octonions do form a ring, as indeed does any algebra over any field. --Zundark (talk) 18:38, 10 May 2009 (UTC)


 * Perhaps I'm confused as to the most common definition of a ring. The requirement that a the ring multiplication be associative has always been encoded in my experience, and that seems to be more or less the standard for the what the thing called "ring".  As I wasn't here for the main discussion, I'm somewhat at a disadvantage as far as knowing what's already been discussed.  I suppose my main question is:  is it known about how much would need modification if the ring definition were adjusted to lack the monoidal requirement on its multiplicative structure? Guardian of Light (talk) 19:50, 11 May 2009 (UTC)

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)

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)

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)

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

another small suggestion
Please add a link to explain "formal symbol".

Also, in the first bullet under "Constructing new Rings from given ones", the statement

The "identity map" from R to Rop is an isomorphism if and only if R is commutative.

is confusing when followed by However, even if R is not commutative, it is still possible for R and Rop to be isomorphic.

It sounds as though the condition isn't "if and only if", but I'm not a mathematician. —Preceding unsigned comment added by 72.221.104.147 (talk) 16:34, 23 August 2008 (UTC)

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))

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" &mdash; Paul August &#9742; 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).)

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:

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 &#9742; 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)


 * In (Allenby, R.B.J.T, Rings, Fields and Groups (first published 1991)) a ring is defined to not necessarily require a multiplicative identity. The term 'Ring with unity' is used if the ring has a multiplicative inverse. Allenby also defines rings to be associative under multiplication. 129.11.126.244 08:42, 23 February 2007 (UTC)

Some examples are not really examples
Some examples seem to be really properties/theorems. For instance: This is not really an example. Maybe this one (and other like this) should be moved to a special 'Theorems' section.
 * A ring (in the categorical sense) is commutative iff it is equal to its opposite ring.

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 &#9742; 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)

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)

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 &#9742; 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 &#9742; 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: 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)
 * Cohn (2002), An Introduction to Ring Theory
 * Rowen (1991), Ring Theory
 * Lam (2001), A First Course in Noncommutative Rings


 * 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)

There are many authors who do not require rings to have unity. For example:
 * Frank Ayres, Jr. (1965), Modern Abstract Algebra
 * Gallian (2006), Contemporary Abstract Algebra
 * Hibbard (1999), Exploring Abstract Algebra with Mathematica
 * Milies (2002), An Introduction to Group Rings (Algebras and Applications, Volume 1) (Algebra and Applications)
 * Giambruno (2006), Groups, Rings and Group Rings (Lecture Notes in Pure and Applied Mathematics)

And according to Wolfram: A ring in the mathematical sense is a set S together with two binary operators + and * (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions:  1. Additive associativity: For all a,b,c in S, (a+b)+c==a+(b+c),  2. Additive commutativity: For all a,b in S, a+b==b+a,  3. Additive identity: There exists an element 0 in S such that for all a in S, 0+a==a+0==a,  4. Additive inverse: For every a in S there exists -a in S such that a+(-a)==(-a)+a==0,  5. Left and right distributivity: For all a,b,c in S, a*(b+c)==(a*b)+(a*c) and (b+c)*a==(b*a)+(c*a),  6. Multiplicative associativity: For all a,b,c in S, (a*b)*c==a*(b*c) (a ring satisfying this property is sometimes explicitly termed an associative ring).  Conditions 1-5 are always required. Though non-associative rings exist, virtually all texts also require condition 6 (Itô 1986, pp. 1369-1372; p. 418; Zwillinger 1995, pp. 141-143; Harris and Stocker 1998; Knuth 2000; Korn and Korn 2000; Bronshtein and Semendyayev 2004).

So rings without the requirement of unity are the standard way that rings are defined by published mathematicians with only a few mathematicians deviating from the norm. Consequently, Wikipedia should get rid of the requirement that rings have a unity and add: "Some authors also require rings to have a multiplicative identity, such a ring is said to be a "ring with unity"". _selfworm _ <font color="#FDD017">( Talk <font color="#FDD017">· <font color="#FDD017">Contribs ) _ 19:05, 21 April 2007 (UTC)
 * I disagree. By and large, rings without unit are the exception in recent works. The most influential English-language graduate algebra textbook (Lang) requires a unit element. So does Jacobson, who uses rng for a ring possibly without unit. Bourbaki requires a unit element, and they are so influential that authors often specifically warn the reader when they deviate from Bourbaki terminology. All algebraic geometers use rings with unit. This can be seen in textbooks such as Hartshorne's Algebraic Geometry, or Mumford's Red Book of Varieties and Schemes. Throughout commutative algebra, unit elements are required, e.g. in Eisenbud's book, Atiyah and Macdonald's book, Matsumura's book, Serre's Algèbre locale et multiplicités. I suppose the essential fact you need a unit for is the existence of maximal ideals. I have personally never encountered a text that omitted the unit requirement without a warning to the reader, whereas this is quite common in texts which require a unit. Joeldl 19:36, 21 April 2007 (UTC)

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)

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)

Alternate definition, or different object altogether?
In Halmos' book on measure theory he defines a ring as a class of sets closed under pairwise unions and differences. It's not immediately obvious to me that this object is equivalent to the concept of a ring presented in this article. Is it? Trevorgoodchild 20:16, 6 March 2007 (UTC)


 * It's not. If you assumed that they were all subsets of a given set, which itself belonged to the set of sets, then that would form a ring (algebraic meaning) by taking addition to be symmetric difference and multiplication to be intersection. You should sign your posts with four tildes. Joeldl 09:28, 17 February 2007 (UTC)

See ring of sets. --Zundark 10:34, 17 February 2007 (UTC)

Options for poll on unit requirement
I think it would be appropriate to hold a poll on the unit requirement. I can see three main options:

1. Unit Use the term ring to mean "ring with unit" by default, and pseudo-ring for "ring possibly without unit". If, in a given article, it is more appropriate to use the "non-unit" definition of ring, do so, but warn the reader.

2. Non-Unit Use the term ring to mean "ring possibly without unit" by default, and unital ring for "ring with unit". If, in a given article, it is more appropriate to use the "unit" definition of ring, do so, but warn the reader.

3. Neutrality Always warn the reader. In articles where it makes little difference which definition is adopted, respect the choice of the first contributor.

Maybe somebody can change Option 3 to something better, I don't know. I'm against Option 3. Joeldl 20:15, 21 April 2007 (UTC)


 * My preference is #1. Everything I've ever read about rings seems to require 1 in the ring. - grubber 21:08, 21 April 2007 (UTC)


 * We are currently on #1 (though we don't usually use the term "pseudo-ring"). If we change to #2 or #3, who's volunteering to go through all the articles that mention rings and change them to match? --Zundark 21:13, 21 April 2007 (UTC)
 * I suppose if we change to 3, it can be done one article at a time without being systematic. If we change to 2, we'll have to start with the basic pages like ring and subring. The work involved is a serious argument against 2 that I'm sure many will be sensitive to. Joeldl 22:41, 21 April 2007 (UTC)

4. Cases If the definition of ring makes a difference in the article, then state what the differences are between the unity and non-unity definitions. If the definition of ring does not make a difference in the article then just say ring without mentioning whether or not the ring has unity. <BR><BR> For 4 There is currently no consensus among published mathematicians about which definition to use; looking at the above comments it is clear that many mathematicians use default unity, while many others do not. Many mathematics will just define ring in whatever way best helps them in their research. In most ring theory math books, authors will explicitly tell the reader whether or not their definition of ring contain unity.<BR>There are some benefits to using ring without a unity as the definition. For instance, with definition two every set that satisfies definition two immediately satisfies definition one, but with definition one there are many sets that would satisfy definition two but not definition one.<BR>I believe that we should use option four since in this way readers who may only be accustomed to one of the definitions will not be lost or confused by the article. Among other other reasons.<font style="border:solid 1px #FDD017; background:#342D7E;" color="#342D7E">_<font color="#FBB117">selfworm _ <font color="#FDD017">( Talk <font color="#FDD017">· <font color="#FDD017">Contribs ) _ 21:31, 21 April 2007 (UTC)

What I meant by "little difference" in Option 3 was that although some details would change, nothing substantial would, and we could leave it to the reader to work it out. It would become extremely burdensome to state all theorems in two forms, even when it was easy to go from one to the other. I have no objection to including Option 4 as a choice, but it seems like a lot of work if we choose that. Do people think that in a poll we should suggest neutral options ourselves or let respondents suggest their own? We could have "Generally favourable to Option 1", "Generally favourable to Option 2" and "Other" sections, without prescribing a neutral option. Joeldl 22:41, 21 April 2007 (UTC)

In practice, people will not always have sources available to them which give statements in both cases, so I think people should be free to choose the definition they prefer for statements they make. However, other editors should have the liberty to change statements to the default definition if there is no substantial reason to prefer the nondefault definition. Also, the legibility of texts will be seriously affected with dual statements everywhere, particularly where the differences are trivial, even if strictly speaking statements may be incorrect for the nondefault option. It is worth mentioning the nondefault option when differences are nontrivial, for example with respect to the existence of maximal ideals. Joeldl 22:57, 21 April 2007 (UTC)


 * Are we discussing options for a poll or actually taking a poll? In any case, I prefer option #1 as I've stated many times before on this page and elsewhere. In tens years time, I wonder how many megabytes of wikipedia will be dedicated to arguing over whether rings should have units or not. Maybe we should setup a dedicated page for it (I jest, of course). -- Fropuff 03:38, 22 April 2007 (UTC)
 * So far we've been discussing options. I would like to settle the issue. Some people take it seriously. I also favour Option 1, but it seems algebraic geometry/number theory people have a different take than ring theory/undergraduate people. (Just kidding — sort of.) If there have been previous polls I'm not aware of, I'd be happy to drop it, but it seems fair the anti-unit people should have their chance. Joeldl 03:56, 22 April 2007 (UTC)
 * I sympathize completely. To my knowledge there has never been a poll on the matter, but it has been discussed plenty of times (here and elsewhere). I would like to see it "settled" too. However, it seems to me that no matter what is decided, a year from now someone not liking the decision will start the discussion up all over again. I'm not suggesting the matter should be dropped, just lamenting over the inevitable. -- Fropuff 04:14, 22 April 2007 (UTC)

Table of publications
Basic algebra books are likely to give full definitions intended for beginners. These are to be distinguished from definitions which appear to be intended purely for disambiguation purposes. For example, if the book gives a definition of ring but not of ideal, it is likely to be for disambiguation purposes. Only books or articles assuming associativity should be mentioned. Joeldl 23:24, 21 April 2007 (UTC)

* Northcott does not include unity in the definition of ring, but notes that rings in the text should be assumed to have unity.

* Smith & Romanowska have definition 1.3.1 "'ring' or 'unital ring' or 'ring-with-1'", and definition 1.3.2 "'ring' or 'non-unital ring' or 'ring-without-1'" and instruct to "determine from context".

I looked up the definition of "ring" in about 20 books in our Mathematics Reading Room. Only one had a unity included (Lang). (There is a notational ambiguity in the table; Herstein and Gallian (and others) use "unit" to mean "an element which has a multiplicative inverse", not the multiplicative identity, which they call "unit element" or "unity", respectively.)

--kundor (talk) 13:51, 8 February 2008 (UTC)


 * I've added a few more just to liven things up. One does notice some interesting trends in the table above. The books which do not require an identity are almost all beginner/undergraduate texts, while the ones that do are almost all advanced/graduate texts. I suspect this is because the most basic properties of rings can be easily formulated without requiring an identity, but in order to do anything serious the existence of an identity becomes essential. -- Fropuff (talk) 07:20, 9 February 2008 (UTC)


 * This is sort of true, but there is a classical exception and a 20th century exception that should caution against taking the coincidence too seriously. A classical exception is the theory of the nilradical and Jacobson radical, which are handled in Jacobson's Basic Algebra and Lam's Theory of NC Rings for non-unital rings perhaps because this was an important motivating case.  A general ring without identity is perhaps not very interesting, but a finite dimensional associative algebra can be quite interesting even without an identity.  Another advanced area where rings without unity are important are the more modern (ok Kronecker considered this too, but Gabriel-Roiter is more recent) representations of quivers; if the quiver is finite, then there is an identity, otherwise there are just finite approximations to the identity.  Finitely supported endomorphisms of infinite dimensional vector spaces are similar (the analysts may want to call them algebras of compact operators); there is no identity, but there are idempotents that approximate it arbitrarily closely.  These examples and others are contained in the book of Wisbauser on categories of modules. JackSchmidt (talk) 08:11, 9 February 2008 (UTC)


 * I looked at many texts in our library, and I also noticed similar trends. My (completely unscientific) impression is that:
 * Introductory undergraduate texts, texts titled "Abstract Algebra" or "Modern Algebra", and American texts are more likely to omit unity.
 * More advanced texts, texts with "Ring Theory" in the title, and European texts are more likely to require unity.
 * --kundor (talk) 19:59, 15 February 2008 (UTC)

It seems that Bourbaki omitted the requirement of a unit in early editions, but that they later reversed themselves. 67.150.247.77 (talk) 17:02, 15 November 2008 (UTC)

Proposed language for poll on unit requirement for rings
It seems it would be best not to try to determine ahead of time the form of the "neutral" options.Joeldl 04:14, 22 April 2007 (UTC)

<div class="boilerplate" style="background-color: #efe; margin: 2em 0 0 0; padding: 0 10px 0 10px; border: 1px dotted #aaa;"> Should Wikipedia favour a default definition of rings: a) requiring a unit element; or b) not requiring a unit element? Currently, the default is to require a unit. (See WikiProject Mathematics/Conventions.) Various options are available as to how to put this into practice, but there seems to be agreement that the default can be overridden if readers are warned explicitly of a departure from the default definition. The choice for rings would likely extend to the definition of ring homomorphisms, subrings, and modules over rings (unital or not).

Comments on proposed poll language
Looks good to me. -- Fropuff 04:27, 22 April 2007 (UTC)
 * Would it be accurate and relevant to say that the current default is with unit? Joeldl 04:34, 22 April 2007 (UTC) That is, accurate, relevant and fair? Joeldl 04:38, 22 April 2007 (UTC)
 * Yes, I think so, per WikiProject Mathematics/Conventions. -- Fropuff 04:44, 22 April 2007 (UTC)
 * Thanks, I've made the change. Joeldl 04:52, 22 April 2007 (UTC)

Since there are no objections, I'll go ahead with this language. Joeldl 23:32, 30 April 2007 (UTC)

Poll on unit requirement
Should Wikipedia favour a default definition of rings: a) requiring a unit element; or b) not requiring a unit element? Currently, the default is to require a unit. (See WikiProject Mathematics/Conventions.) Various options are available as to how to put this into practice, but there seems to be agreement that the default can be overridden if readers are warned explicitly of a departure from the default definition. The choice for rings would likely extend to the definition of ring homomorphisms, subrings, and modules over rings (unital or not).

Generally in favour of definition with unit

 * Support unit default By far the majority of sources I've seen, including Bourbaki, require a unit element. In the areas of commutative algebra and algebraic geometry at least, this convention is universal, and I do not believe a warning is even called for. Not necessarily unital rings should be called pseudo-rings (that is the Bourbaki term, isn't it?), unless a specific convention to the contrary is mentioned in an article. Joeldl 23:41, 30 April 2007 (UTC)
 * Support per Joeldl. -- Fropuff 04:14, 1 May 2007 (UTC)
 * Support with reservation: it's not necessary to invent special terminology for non-unital rings and algebras; rings without unit do occur naturally in mathematics, e.g. as rings of operators, consequently, in K-theory and non-commutative geometry, and it would be prudent to mention this fact.Arcfrk 04:40, 1 May 2007 (UTC)
 * As the preface to the poll says, there is nothing preventing an article from adopting the non-unit definition of ring, provided it warns the reader. I don't think using the Bourbaki term is "inventing terminology". There are a profusion of terms in existence, and it makes sense to adopt the Bourbaki one. There is a possibility that in an article both definitions of "ring" will appear, so there should be a word for the non-unital ones. Joeldl 06:35, 1 May 2007 (UTC)
 * Bourbaki did invent a lot of terminology (and notation, of course). That's what makes reading Éléments de mathématique so trying. Some of their inventions were widely adapted, some weren't. Pseudo-ring is so obscure that unless you want to insist that everyone goes back and carefully reviews the Bourbaki definition whenever it's used, it should better not be included. Arcfrk 19:10, 1 May 2007 (UTC)
 * I think reading Bourbaki is difficult because mathematics is difficult. Joeldl 22:27, 2 May 2007 (UTC)


 * Support. This is the status quo and doesn't present real difficulties. Charles Matthews 14:55, 1 May 2007 (UTC)
 * Support Harmless; any unitless ring can be trivially embedded in one with unit. Jacobson's term for a unitless ring is "rng" (no i, you see), and IIRC he borrowed it. Any effort to make this fundamentally unimportant distinction should make clear what they mean at the point the distinction is made, not rely on a link; and if we do that, we don't need to worry about uniform nomenclature. Septentrionalis PMAnderson 21:00, 2 May 2007 (UTC)
 * Support. This is the current Wikipedia convention, and the most common convention in the literature. There is no reason to consider changing. --Zundark 14:14, 3 May 2007 (UTC)

Generally in favour of definition without unit

 * Support, per "reservation" part of Arcfrk's comment above. Mct mht 04:54, 1 May 2007 (UTC)


 * Support By now the poll is so old that maybe there seems little point in responding, but I should like to express my view. I am very surprised by the whole discussion. I had always thought that the definition without requiring a unit was standard. I am aware that frequently the definition with a unit is used, but I had always thought this was because if you are writing about rings with unit then it is cumbersome to keep having to write "ring with unit" . . . "ring with unit" . . ., and so you say once and for all "for the purpose of this book/article, 'Ring' will be taken to mean 'Ring with unit". It had never occurred to me that there was a large body of mathematicians who regarded that as the default definition. Incidentally, quoting Bourbaki as an authority is odd, as Bourbaki's usage is so often perverse that they can scarcely be taken as representing the norm. JamesBWatson (talk) 13:23, 2 December 2008 (UTC)

Discussion

 * Object. Why are we having a poll? What can this accomplish? Wikipedia mathematics articles cover a broad range of topics, and most when requiring a ring also require an identity, but some do not. We already have a stated convention of assuming an identity, but a reader should never be left wondering. Awkward as it is, we do not have the luxury of most book authors, to choose one definition to cover our focused needs. A reader of a given article cannot be expected to know — nor easily find — our assumptions, nor can an editor; they bring their own assumptions! We should address this challenge, which a poll on ring identities does not do. --KSmrqT 05:59, 1 May 2007 (UTC)
 * Probably very little. But if you skim through this talk page you'll see that a number of users have asked on a number of occasions that we change our definition of a ring to not require an identity. If the current definition is to stand that it should have the broad support of mathematics editors. Perhaps a poll can show this, or perhaps not. But it certainly does no harm. -- Fropuff 06:21, 1 May 2007 (UTC)
 * Presumably, if an article refers to rings, there will be a link to ring (mathematics), and the reader can assume that it is the definition there that is used. I cannot imagine that many non-unit people (and I'm thinking of mathematicians) would be unaware of the predominance of the unital definition (Bourbaki, etc.), so they will naturally pay attention to determine which definition is used. It is not a reflex for most unit people to consider the possibility of non-unital rings. Somebody pointed out earlier that Lam's book on Noncommutative rings has the unital convention. I do not believe that we are dealing with conventions of equal status. Joeldl 07:06, 1 May 2007 (UTC)
 * To the contrary. I am an undergraduate student of mathematics, and until coming across this article had never encountered the definition of ring requiring a unity.  I had Joseph Gallian's Contemporary Abstract Algebra and I.N. Herstein's Topics in Algebra as relevant texts, and all my professors have always assumed rings lacked unity in general (without mentioning contrary conventions.)  In fact, one professor based much of his development of ring theory on the lack of unity.
 * So it would be accurate to say that I was completely unaware of the "predominance" of this unital definition. Moreover, any student coming out of this department would be the same way, and I cannot imagine that Miami University is isolated among American institutions. --kundor (talk) 14:01, 8 February 2008 (UTC)


 * I fully agree with KSmrq. We cannot afford to depend on a convention - it shoudl be made clear to the readers of each article. JPD (talk) 10:07, 1 May 2007 (UTC)
 * That depends on one's assessment of the predominance of one convention over the other. As I said, I don't believe anyone who uses the non-unital convention is going to approach a text naïvely expecting that others will follow that convention. If you know of publications that go one way or the other, you can add them to the table above so we can get an idea of how the conventions compare. Joeldl 11:43, 1 May 2007 (UTC)
 * I really don't see the thing about not being able to afford conventions. Mathematical terminology changes over time, and we cannot expect to have totally agreed, uniform terminology. But we can have defualt conventions. Every graph theory lecture used to have to start with a statement of conventions, but I believe "loopless, no multiple edges" became the default convention. That saved everybody some time. Charles Matthews 15:03, 1 May 2007 (UTC)


 * How about creating a page ring (nonunital) with the non unit definition. Then if an article requires the other definition it can link there. --Salix alba (talk) 10:56, 1 May 2007 (UTC)
 * I don't think the two notions are so different that they aren't covered adequately in this one article, especially since it is possible to link to a section of an article. The question is whether how much we depend on conventions for convenience in other articles. The convention we do have already leans the way that most people are suggesting, but our conventions cannot be used as strongly as they would be in a book, and should not be an excuse for not making each article as clear as possible. JPD (talk) 14:41, 1 May 2007 (UTC)


 * Much as we might wish we had the power to decide this (and while we are at it decide whether the natural numbers include 0 or not) we don't. Standard books disagree.  All we can do is report what standard books on the subject say. Rick Norwood 13:07, 3 May 2007 (UTC)
 * Wikipedia should report, in the article Ring (mathemetics), what conventions exist in the literature. How Wikipedia itself uses the word ring is a matter for Wikipedia to decide, no doubt taking into consideration the existence of multiple conventions, but also their relative prevalence and practical issues concerning the organization of the encyclopedia's content. Joeldl 01:14, 4 May 2007 (UTC)

Definition not precise?
The definition of ring now reads 'A ring is a set R equipped with two binary operations +: R + R -> R and x: R x R -> R (where x denotes the Cartesian product), called addition and multiplication, such that...'. It looks to me as if 'x' is being used with two different meanings here, and the same for '+'. Should not the text read: 'A ring is a set R equipped with two binary operations +: R x R -> R and *: R x R -> R (where x denotes the Cartesian product), called addition and multiplication, such that...'? --Sergut 18:08, 3 December 2007 (UTC)


 * Thanks for pointing this out. It was correct until about a week ago, when someone messed it up. I've fixed it now. --Zundark 19:30, 3 December 2007 (UTC)

Question: Center of the ring of all square n-by-n matrices with entries from R
This center surely contains all diagonal n-byn matrices. Does it contain any other elements? TomyDuby (talk) 04:15, 20 July 2008 (UTC)


 * this ring has trivial center: r&middot;1 where 1 is the identity matrix and r a number. Mct mht (talk) 05:53, 20 July 2008 (UTC)


 * To be clear: The diagonal matrix [1,0;0,0] is not in the center.  For any ring R with 1, the center of the ring of n×n matrices is the set { r·I : r in Z(R) } of all scalar matrices, where I is the identity matrix and Z(R) is the center of R.  To see this, it suffices to check commutativity with the matrices which have exactly one nonzero entry. JackSchmidt (talk) 11:26, 20 July 2008 (UTC)

The purpose of a Wikipedia talk page is to discuss possible edits to the article, not to serve as a general discussion forum on the subject of the article. Besides, the query has been answered. If nobody raises any objection I shall delete this section.JamesBWatson (talk) 13:33, 2 December 2008 (UTC)
 * Perhaps move it to Talk:Center (algebra) instead.Rgdboer (talk) 22:09, 2 December 2008 (UTC)

Assessment comment
Substituted at 21:16, 4 May 2016 (UTC)