# Talk:Noetherian ring

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Field: Algebra

## Chain vs. Set

It seems to me that the sentence "In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring in which every non-empty set of ideals has a maximal element."

"In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring in which every non-empty chain of ideals has a maximal element." Kerry (talk) 01:10, 5 August 2014 (UTC)

You are essentially saying "we should define it using the ascending chain condition rather than the maximum condition," but no reason is given why. In ordinary context (ZFC) the two are equivalent, so there is nothing wrong here. Do you have a special reason you suggested this? Rschwieb (talk) 12:56, 4 August 2014 (UTC)
I see your point, but wouldn't it be better to use a definition that doesn't assume The Axiom of Choice unless technically necessary? That's my motive for preferring the ACC rather than the maximal element approach. Kerry (talk) 01:09, 5 August 2014 (UTC)
I can only think of two things to say.
A) Policy says we should avoid putting too much emphasis on mathematics done without the AoC: the lion's share of ring theory is done in the context of ZFC.
B) The chain condition version is the weaker of the two definitions. Even if we were concerned about providing adequate definitions outside the use of AoC, I'm not sure if the chain condition is "the right" version of Noetherianness to use. The ACC is a nice simplification of the maximum condition afforded to us by the AoC.
This being the case, there does not seem to be any real reason to make a change. Rschwieb (talk) 12:25, 5 August 2014 (UTC)
Mathematically nothing is wrong, but I do wonder the choice of the definition seems a bit arbitrary. (On somehow related we should add an example of the use of this particular definition, like a proof that involves annihilator and associated prime.) -- Taku (talk) 13:36, 5 August 2014 (UTC)

## Ideals of Q

The article states that noetherian implies every ideal is finitely generated; shouldn't that be every proper ideal is finitely generated, since for example, Q is noetherian (every field is), and yet Q cannot be finitely generated? Chas zzz brown 20:11 Apr 2, 2003 (UTC)

Never mind. Q is finitely generated - as an ideal. Chas zzz brown 10:22 Apr 3, 2003 (UTC)

## Large

The article says: "Rings that are not Noetherian tend to be (in some sense) very large." I dont know what this should mean, since there are many examples of non-noetherian subrings of (not very large) noetherian rings. For example, there are non-noetherian subrings of k[X,Y]. ( k[X,Y] the ring of polynomials in two variables over a field k) 84.154.252.105 14:43, 8 April 2007 (UTC)

I added two examples to ensure that the "large" comment is not taken too formally. I suspect all k-subalgebras of k[x,y] itself are noetherian, but I gave the reasonably standard example of a non-noetherian subring of k(x,y) that is a non-finitely generated module over k[x,y]. JackSchmidt (talk) 15:13, 22 May 2009 (UTC)

## lower case 'n'?

This should be discussed in the article, but shouldn't "n" in "Noetherian" be lower-cased, just like abelian. -- Taku (talk) 23:02, 19 December 2008 (UTC)

Use of lower case for adjectives named after mathematicians is commonplace in French, but is all but unheard of in English. The word abelian is a notable exception to this rule. (I have never heard a convincing explanation for why this one word is not capitalized, but it invariably isn't.) Plclark (talk) 06:58, 24 December 2008 (UTC)
It is not uncommon for noetherian and artinian to be lower case in mathematical English, but less common than for abelian. I don't think it's worth changing one way or the other. Another silly example is whether the φ in Frattini subgroup should be capitalized; it usually is, but not always. Mathematical English is a very international language, and it shouldn't be surprising that its rules are complex and somewhat incomprehensible. JackSchmidt (talk) 18:08, 24 December 2008 (UTC)
The common convention is to use lower case on "noetherian", as she was a woman. —Preceding unsigned comment added by 138.246.7.139 (talk) 09:47, 22 May 2009 (UTC)
That is, for obvious reasons, an unconvincing argument. PST 11:06, 23 May 2010 (UTC)

## Dedekind domains

There really should be something about Dedekind rings in this article since the "typical proof" that non-zero ideals in Dedekind rings can be uniquely factorized into a product of powers of prime ideals (roughly speaking) relies on a couple of basic facts of Noetherian rings. Also, Dedekind rings are ubiquituous in number theory and commutative algebra. I suspect Plclark is an expert on these matters as he seems to have contributed to the article Dedekind domain, but if no-one adds something about Dedekind domains here, I can always do so. PST 01:47, 7 August 2010 (UTC)

## DCC on prime ideals

"...if a ring is Noetherian, then it satisfies the descending chain condition on prime ideals." Is this really true? — Preceding unsigned comment added by UqSL2 (talkcontribs) 22:02, 25 January 2012 (UTC)

## Long "introduction" section

A while back User:TakuyaMurata removed a long expository section which User:Point-set topologist recently reinstated. I reverted to restore Taku's version because I think his earlier removal was probably the right way to go. As it was, it was a bit long and apparently springs out of personal thoughts, and wasn't citable and wasn't really encyclopedic. It also served as a fairly big text barrier against the main information in the article in the next section. I'm sure some of the information thta used to be there can find a home elsewhere in the article, though. Rschwieb (talk) 18:17, 31 July 2013 (UTC)

My position is perhaps clear, but I will restate it anyway. I admit my edit was not constructive, and I admit that the article of this sort needs a "good" introduction: I image a significant percentage of the readers will be undergraduates, and they should be given something less dry than the type of the exposition that math graduate students prefer. This makes a good case for the "good" introduction. My problem is that the version I removed is not "good" in the sense that, for example, it is not neutral (as Rschwieb pointed out) and also is misleading in some parts; for example, that a PID is Noetherian is rather trivial. I also don't understand the point of emphasizing Z. The main point of "Noetherian ring" are twofold: (i) virtually every ring that arises "in nature" turns out to be Noetherian and (ii) the ring theory is much nicer for Noetherian rings. It is indeed quite amazing that, as it turned out, the ascending chain condition is exactly the condition that leads to so many deep theorems. I mean how can you just come up with the idea of ACC from nowhere? (of course, it's not from nowhere.) In my mind, the "good introduction" thus emphasizes applications and the ring theory for Noetherian rings, without getting into too much details. It would be "very nice" if you can explain why ACC turned out to be the right condition for so many nice theorems, but that's probably a hard problem that we don't need to attack here. (I just didn't think the materials I removed succeeded in unlocking the mystery.) -- Taku (talk) 00:17, 1 August 2013 (UTC) P.S. I also imagine some in-negligible percentage of readers will have the "non-algebra background" (e.g., they specialize in analysis) and the introduction should also be interesting to those segments of the readers. -- Taku (talk)

Hi User:Rschwieb and User:TakuyaMurata, I see both of your points but I do think that until one of us can propose a better introduction, the introduction that I reinstated should stay. It's true that there are more important things to be said and that it's a little too long. But I think it's a good motivation for students learning the material for the first time. So, I'm reinstating the Introduction for now. (Let me emphasise that while I do think that there could be a better introduction, it seems to me more constructive to have at least one introduction than none at all. Taku's points are also excellent.)

A shorter version successfully describing the basic nature of Noetherian rings would be good. But it should be conveyed without descending into a homegrown exposition about them. I'm in agreement with Taku's description of a "good introduction" in the last paragraph. Rschwieb (talk) 12:51, 2 August 2013 (UTC)