Talk:Ring (mathematics)

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Ring (mathematics)[edit]

A more basic way to learn rings, beyond comparison to modulus, is to investigate Vector space "Definition" (Linear algebra topic) which is ring for simpler objects (linear) and which shows readily hand done ways to determine or create.

Does an ideal have to be defined as an additive subgroup?[edit]

The definition currently in the text is "A subset I of R is then said to be a left ideal in R if R I \subseteq I." I suggested the definition should begin "An additive subgroup I ..." I appreciate that if R is a ring with 1, as is assumed in the article, then the definition as stated implies that I will be an additive subgroup in particular. However, I note that of the texts immediately available to me, Herstein, Jacobson, Lang, Lewis and van der Waerden all include the requirement of being an additive subgroup in their definition of ideal. So perhaps my suggestion is consistent with the reliable sources. Deltahedron (talk) 11:38, 21 September 2014 (UTC)

As the text contains the definition of RI, I agree that "additive subgroup" is not formally needed. However, it may be useful, as, without it, the definition is confusing for those who omit to read the definition of RI. On the other hand the present definition is unnecessarily too technical. I suggest "A subset I of R is then said to be a left ideal in R if the sum of two elements of I and the product of an element of R and an element of I always belong to I. Equivalently, I is a left ideal if RI\subseteq I, whereR I denotes the span of I over R, which is the set of finite sumsr_1 x_1 + \cdots + r_n x_n, \quad r_i \in R, \quad x_i \in I. This isequivalent to say that I is a left submodule of R (viewed as a module over itself)." The definition as submodule is lacking and is fundamental in many applications. By the way, a section on modules is lacking in this article (The section "The action of a ring on an abelian group" may hardly be considered as a section on modules). D.Lazard (talk) 12:33, 21 September 2014 (UTC)
I now see that maybe the "ideal" (couldn't resist) definition of an ideal might be something like that of subring: a subset I is an ideal if x + y and rx are in I for all x, y in I and r in R. The parallelism is helpful for the readers to compare the two definitions. The definition "RI\subseteq I" is because it really explains the only requirement is that R acts on I as a ring; i.e., I is a submodule. I'm not too sure about the "module section"; obviously, the readers should go to module (mathematics) to read about modules. There is some ring-theoretic aspects of the modules theory (hence, the emphasis on ring action), but the article shouldn't get too deep into modules, in my opinion. -- Taku (talk) 14:33, 21 September 2014 (UTC)
I went ahead and put it back to including the "additive subgroup" because that is the most common wording. We gain transparency, and any parallelism with other definitions is a bonus. Rschwieb (talk) 13:01, 22 September 2014 (UTC)

Redundancy of ring axioms[edit]

Interestingly, it is not circular to conclude that commutativity is a consequence of the other axioms. One could probably drop some other axiom (say, right distributivity) instead. That this redundancy is not obvious (as seen by the to-and-fro) suggests that it makes sense to note that the ring axioms are redundant, but singling out commutativity alone for such treatment may not be sensible. —Quondum 05:15, 23 April 2015 (UTC)

In my textbooks, commutativity of addition is defined before the axioms of multiplication are defined and before cancellation laws are deduced. It did not make sense to me to say that commutativity of addition follows from the other axioms. If it can be shown that an axiom can be deduced from axioms which follow it, I think that should be stated in a separate section. I think it should also be proven rigorously, since sometimes an axiom only appears to follow from others. For example, the reflexive property of an equivalence relation. — Anita5192 (talk) 05:35, 23 April 2015 (UTC)
I have reinserted it under the guise of general redundancy of axioms, with a clearer, simpler proof. I think that it would be inappropriate to single out commutativity as the redundant axiom, but any of several could presumably be omitted. It may be seen that both distributivity laws, multiplicative identity and cancellation all are used. In this guise, whether the omitted axiom is somewhere in the middle hardly matters. But the sheer surprise we all seem to experience when seeing this seems to argue that a note in the article is warranted. Do you think I've met the criteria you've mentioned (separate section, rigorous proof, also made much clearer)? —Quondum 05:53, 23 April 2015 (UTC)
PS: My favourite example of redundancy of axioms is for a group. We can define a group as a nonempty associative quasigroup. Note that there is no need to axiomatize an identity or inverses; two-sided division suffices. —Quondum 06:06, 23 April 2015 (UTC)
I don’t think Slawekb’s last two edits were helpful, as the fact that commutativity of addition follows from the other axioms is interesting, unlike the trivial fact that one half of the additive inverse axiom is redundant in the presence of commutativity (not to mention that the choice of axioms is now weirdly inconsistent, as the additive identity axiom is still stated as two-sided). It’s not OR. Some references can be found in http://math.stackexchange.com/a/346682 , for instance.—Emil J. 11:30, 23 April 2015 (UTC)
By the way, the same argument also shows that commutativity is redundant in the axioms of modules (and vector spaces).—Emil J. 11:43, 23 April 2015 (UTC)
The axioms we give at vector spaces only assume that the additive inverse and identity are one-sided. So, no, it does not follow that commutativity is redundant with the other axioms. Sławomir Biały (talk) 11:51, 23 April 2015 (UTC)
You know perfectly well what I meant, and anyway, this is of no concern to this article.—Emil J. 11:54, 23 April 2015 (UTC)
I don't think the adversarial tone is constructive. I don't "know perfectly well what [you] meant", and if it "is of no concern to this article", why bring it up in the first place? Standard sources define rings by three axioms, not eight. So it makes sense to write the article from that perspective. None of these sources point out any redundancy in the axioms and, indeed, these three axioms are manifestly independent of each other. (Although this does not preclude the possibility of axiomatizing a ring in some other way; naturally that would require a good source to include.) The source mentioned at stackexchange does not directly support the claim made in the article. Betsch is discussing fields rather than rings, and the statement lacks a proper citation anyway. Sławomir Biały (talk) 12:16, 23 April 2015 (UTC)
Well, well, well. Much ado about nothing. I notice that no-one has addressed my opening comments in this thread. I really don't see why a separate note about axioms being redundant should trigger a flurry of edits on the axioms themselves. There is nothing wrong with axioms being redundant; this is quite normal, as the note mentioned. And if I was to go through this and other maths articles removing everything that is unsourced, I'm afraid the content would be sorely diminished; I would be interested in comment on the consistency of the application of criteria for inclusion. —Quondum 14:22, 23 April 2015 (UTC)
It's not for want of looking that I removed the content in question. Indeed, all sources that described rings by means of axioms included only three axioms. No comment was made in the standard places that the axioms are not independent, because this depends on the way the three axioms are unwrapped into properties. Naturally, the "properties" that a rimg satisies are not generally expecyed to be independent. No, we shouldn't remove all unrefined content, but when content fails the basic test of WP:V, it should be removed, in my opinion. In this case, it is truly much ado about nothing. There is not even the barest indication that these issues have been considered anywhere in the literature. (The nest peopke have come up with is the ضaforementioned stackexchange thread refers to an unnamed paper by Henkel, decades before the concept of a ring was introduces by David Hilbert. Needless to say, this would make an extremely poor source.) Sławomir Biały (talk) 15:33, 23 April 2015 (UTC)
Not “nest”, just first. I. M. Isaacs, Algebra: A Graduate Course, AMS, 1994, p. 160. K. D. Joshi, Foundations of Discrete Mathematics, New Age International, 1989, Exercise VI.1.1, p. 405. For modules: S. Mac Lane, G. Birkhoff, Algebra, AMS, 1999, Exercise V.1.6, p. 162. I’m sure you could find more with a bit of effort, this is simply a well known observation.—Emil J. 16:24, 23 April 2015 (UTC)

"Abelian groups" and "monoids" are too technical[edit]

This article, and in particular the definition of a ring in this article, should be accessible to a person who does not know what an abelian group is, or what a monoid is. An relevant Wikipedia guideline is Wikipedia:Make technical articles understandable. Regarding where we should be on the scale of understandable vs. technical, I think it's a bad idea to follow Bourbaki. By worrying too much about redundancy I think we are at risk of making this article less understandable. I believe introductory textbooks more commonly list in the neighbourhood of 8 axioms, rather than just 3; probably because the 8 easy things are easier to understand than the 3 more difficult things. So I think we should be numbering the axioms as they were before. Mark MacD (talk) 13:27, 23 April 2015 (UTC)

We are following not just Bourbaki, but also MacLane and Birkhoff, and Serge Lang. I note that in the previous version, the words monoid and abelian group also appeared. All I have done is to delineate more clearly that there are only three axioms, not eight. This is well sourced to standard references. I did not see any introductory treatment that called the eight previously numbered items "ring axioms". And indeed it seems like our own sloppiness here is the entire source of the confusion in the previous thread.
Sławomir Biały (talk) 15:20, 23 April 2015 (UTC)
Is it correct to call the three listed criteria "axioms"? Axioms that are defined in terms of more axioms seem to be a misnomer. —Quondum 16:07, 23 April 2015 (UTC)
Yes, they are axioms. See the cited sources. (And anyway, this is a complaint that could just as well apply to the earlier revision, which relies on the notion of "set", which satisfies some axioms of its own that we have not included.) Sławomir Biały (talk) 16:27, 23 April 2015 (UTC)
It was not a "complaint". It was simply a question about terminology. —Quondum 17:04, 23 April 2015 (UTC)
When I searched Google books for "Ring axioms", the first 10 hits all listed between 7 and 9 axioms for a ring (which I could see from the snippet view), numbered in a wide range of different ways. My point is that sources differ regarding the how to define a ring, and even when they agree they differ on how to present the definition; recall this epic table in this talk page's archive. So we should be using our editorial judgement for what is best for the reader, rather than simply following a few favourite sources of mathematicians. Mark MacD (talk) 22:39, 23 April 2015 (UTC)

While they are really definitions, many standard sources say axioms.Rick Norwood (talk) 19:57, 23 April 2015 (UTC)