Talk:Rng (algebra)

Pronunciation?

Please someone explain how to pronounce rng. A suggestion how this translates to other languages would also be interesting. --Zinoviev (talk) 19:51, 22 November 2006 (UTC)

I doubt it is a true English word that can be pronounced. I would simply say "ring without identity". -Wshun (talk) 10:04, 8 February 2007 (UTC)

The current version of the article states: "pronounced rung" — which raises the question how "rung" is pronounced. Could someone provide the pronunciation in IPA please? --132.199.99.50 (talk) 09:59, 17 March 2017 (UTC)

Pronunciation is important as well for a correct article

If it is pronounced as "ar-en-dgi", then "an" before it needed, not "a". — Preceding unsigned comment added by 85.64.219.145 (talk) 10:23, 21 April 2024 (UTC)

Read the lead and you will see rng is pronounced as rung.—Anita5192 (talk) 12:53, 21 April 2024 (UTC)

Thanks. 85.64.219.145 (talk) 15:12, 22 April 2024 (UTC)

Origin (and pronounciation)

Who originated the term "rng"?

And when?

Hazelmaye (talk) 12:38, 27 July 2009 (UTC)

In his seminal work "Basic Algebra", Nathan Jacobson dedicates a chapter to Rngs. He attributes the term to Louis Rowen, and also suggests a pronunciation. 94.159.132.98 (talk) 18:07, 6 November 2010 (UTC)

For the record: The attribution of the term to Louis Rowen is in a footnote on page 155 in Basic Algebra I by Jacobson. The suggested pronunciation in that footnote is: rǔngs (for the plural written rngs).

Louis told me personally (a few years ago; what follows is my recollection of what he told me) how it happened: He walked one day into Jacobson's office. Jacobson asked him what he would call a ring without 1, and he instantly replied: "A rng, of course". Hadaso (talk) 10:00, 13 October 2015 (UTC)

Moved

The content of this article has been moved to rng (algebra). Ebony Jackson (talk) 06:16, 31 December 2013 (UTC)

Splitting hairs

I know that this is arguing against typical use of terminology, but bear with me. Some terms are properties of elements (in relation to the operations), and do not depend on any choice of containing set. For example, whether an element is "an idempotent element" is independent of which subrng (or even subset) of a larger rng is considered. The term "non-zero-divisor", on the other hand, has such a dependence. If P is a subrng of Q and x is an element of P, to say that an element x is a non-zero-divisor is ill-defined. In P it can be a non-zero-divisor, while in Q it can be a zero divisor. The mere mention of a super-rng Q of P (or lack thereof) is not sufficient, even though it is common to have the reader infer which set is meant as the context by the very woolly rules of usual English inference (e.g. "I went to restaurant and I had a meal" would lead to the inference that the meal was had at that restaurant after I arrived, but logically that is not implied). Since we should not assume that readers are as attuned to the routine assumptions and inferences as a mathematician familiar with the terms would be, I feel that a more direct hint of the context in such cases is appropriate.

This instance fairly clearly has such an ambiguity: it is not entirely clear whether the non-zero-divisor is with respect to the image of f (which is itself a rng and the last-mentioned object) or to S. The latter is the intended meaning, because otherwise the statement does no hold. I had the qualification "non-zero-divisor on S", but this was removed (some might have preferred "non-zero-divisor in [or of] S"), so rather than simply reinserting it, and to give scope for a fuller discussion of this subtlety, I am first mentioning this here. —Quondum 21:29, 13 December 2020 (UTC)

Whoops – I omitted what I was referring to by "this instance": I mean the phrase "and the image of f contains a non-zero-divisor". —Quondum 21:32, 13 December 2020 (UTC)

I was the one who removed it. You are certainly right that the notion of non-zero-divisor depends on the parent ring, so it is more precise to specify it explicitly. I removed it only because it seemed clear from context, but maybe it is not as clear as I thought, so I will put it back. I do think that "of" is the right preposition here, however - an element "on" a ring sounds funny to me! Ebony Jackson (talk) 22:40, 13 December 2020 (UTC)

Thanks. It is best to choose language that sounds natural, so that works for me. I am a lot happier with what we have arrived at for that statement: now it is concise, understandable, more informative than it was originally and (I think) relatively free of gotchas. —Quondum 00:30, 14 December 2020 (UTC)

Splitting more hairs

The statement

"If f : R → S is a rng homomorphism from a ring to a rng, and the image of f contains a non-zero-divisor of S, then S is a ring, and f is a ring homomorphism."

presupposes that the canonical assignment of the nullary operation of a ring will be made. In this context, it is helpful to avoid conflating "an element that acts on the set as an identity" with "the image of the nullary operation that satisfies the identities of a multiplicative identity element". (This is the misconception I referred to when I reverted my example: if the structure is maintained, it is not a ring at all despite having a two-sided identity, and partially reassigning structure makes no sense.) In the same vein, even though the assignment is canonical, does it not make sense to say "... then the structure of S can be augmented to make it a ring, in which case f is a ring homomorphism" or "... then S canonically has the structure of a ring, and f is a ring homomorphism"? This feels verbose, but you'll understand my point. —Quondum 14:34, 5 January 2021 (UTC)

I'm not sure I understand what you are saying, but let me try to say something in case it answers your concern. There is more than one approach to the definition of ring. In one approach, the 1 is a nullary operation, which is an additional part of the structure. In the other approach (which is the more common one, and which is the one currently given in the Ring article), the ring is simply the set with the two operations and it is a condition that there exists an element that satisfies the axiom of a multiplicative identity. In the second interpretation, if one has a rng, it either is a ring already or it is not - there is no need to turn the triple (R,+,⋅) into a 4-tuple or anything like that. Ebony Jackson (talk) 17:26, 5 January 2021 (UTC)

I think you have what I was saying. Depending on whether one takes an axiomatic or an equational (a variety of universal algebra) approach, the statement may be fine or need adjustment. The issue I have with the "there must exist an element that behaves like an identity" axiomatic approach in this context is that it does not sit well with the definition of a subring [alternatively ring homomorphism]. One needs to introduce what amounts to an artificial/unusual constraint into the definition. Most other classes of object that I can think of (group, semigroup, vector space, rng, ...) get by on "a subobject is a subset that is closed under and preserves the defined operations". Rings (and monoids) do too if the identity is part of the defined structure (such as a nullary operation), but not if it is just any element that meets the requirements of an identity in the subset. —Quondum 19:39, 5 January 2021 (UTC)

Yes, I think you have discovered for yourself what it took mathematicians many years to realize, that for many purposes it is better if things like the identity and the negation map are interpreted as additional structure instead of just saying that they exist. Once one has that point of view, the definitions of subring, etc., become natural, and there are other reasons too. But still it is more common in most of the mathematical literature to define groups and rings in the other way, with axioms involving existential quantifiers, probably because at first it seems simpler to have to specify fewer operations. Ebony Jackson (talk) 22:03, 5 January 2021 (UTC)

Okay. Useful generalization is something that mathematicians do, and I like. I guess that in this instance I am splitting the hairs too finely: one should allow for descriptions from any current mainstream perspective. ( Speaking of quantifiers, I have yet to convince anyone that the empty set qualifies as a group in universal algebra. :-/ ) —Quondum 22:25, 5 January 2021 (UTC)

Add inline citation

I propose to add an inline citation to Jacobson for 4 reasons 1) he describes the origin, 2) he describes the pronunciation of rng, 3) his book and section on this topic is available to everybody online and 4) Wikipedia wants us to cite sources. This is how it would appear.

The term rng (IPA: /rʊŋ/) is meant to suggest that it is a ring without i, that is, without the requirement for an identity element.

• Jacobson, Nathan (1989). Basic algebra (2nd ed.). New York: W.H. Freeman. ISBN 0-7167-1480-9.

TMM53 (talk) 04:07, 23 March 2023 (UTC)

I added the reference and footnote. TMM53 (talk) 20:48, 2 May 2023 (UTC)

References