Talk:Ring (mathematics)

From Academic Kids

Contents

1 Associativity and unit? 2 Possible Definition Contradiction 3 Definition of ring and unit elements 4 Striving for conventions 5 References for definitions 6 Article is FULL of errors 7 Definition: some references, remove unit element?

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)

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)

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" — 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))

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)