Talk:Ideal (ring theory)

Untitled

I took this out for now:

Alternatively, all of the requirements may be replaced by the following: any finite R-linear combination of elements of I belongs to I

I don't particularly like this, because it hides too many things. For instance, we would have to have the understanding that linear combinations are two-sided linear combinations, whereas in the typical vector space and module setting they are only left-sided combinations (and two-sided linear combinations don't make sense). Furthermore, we would have to spend a paragraph explaining that finite combinations include combinations of zero elements which are defined to be the zero element of the ring that the zero elements were chosen from. In summary, I think this alternative definition looks cuter than it is. AxelBoldt, Sunday, June 9, 2002

It's not meant to be cute; it's meant to show that there is a single broad class of operations — the linear combinations — that ideals are closed under. I know that thinking in these terms makes ideals (and, more generally, modules) clearer to me, but I agree that it's a less elementary point of view. I would be happy to move the comment to the paragraph that mentions the relationship between ideals and submodules. Since linear combinations are inherently central to module theory (that is, linear algebra), this is an appropriate place; additionally, this comes after we've discussed the various flavours of ideals, so that a quick parenthetical "(where the linear combinations are on the left, on the right, or two-sided, accordingly as the ideal)" will take care of that. In any case, I think that it's worth mentioning somewhere, even if way at the bottom; the same thing on the pages Submodule and Vector_subspace (or Module (mathematics) and Linear_algebra/Subspace, which is where those topics are hiding out now). As for the zero linear combination, that can be mentioned on the page Linear_combination (once it exists — I was shocked to see a red link in my Preview!). After all, if anybody is confused about how closure under linear combinations could yield the zero element, then that's what they'd look up, right? — Toby Bartels, Tuesday, June 11, 2002

PS: Hey, no more red link! Needs work, however. — Toby

I'm happy now if you are. The next step is to work on Linear_combination ^_^. — Toby Bartels, Tuesday, June 11, 2002

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I think this page should be moved to ideal (algebra). I can't help but suspect it was written back in the days when "mathematical group" rather than "group (mathematics)" was considered an acceptable title. A lot of links will have to be fixed, but not as many as in the case of the group theory article. Michael Hardy 02:56, 28 Aug 2003 (UTC)

It would be better at Ideal (ring theory), since there are other types of ideals in other sorts of algebras, which don't share most of the properties of ring ideals (except a few basic ones!). To be sure, these are more obscure, so Ideal (algebra) can redirect to the ring theory article until some more general article is written. But we should avoid linking to the more general title, even if it seems safe so far. -- Toby Bartels 15:17, 28 Sep 2003 (UTC)

I'll gladly move this to Ideal (ring theory), if no one objects. Waltpohl 23:26, 24 Feb 2004 (UTC)

Is the trivial ideal proper?

From the article:

We call I a proper ideal if it is a proper subset of R.

When I took algebra, a proper ideal also had to be different from {0}. --Trovatore 04:36, 21 September 2005 (UTC)

'Godhaber and Ehrlich, Algebra (1970) and Clark, Elements of Abstract Algebra (1971), both define proper ideal as an ideal different from {0} and the whole ring. Herstein, Topics in Algebra (1964) and McCoy, Rings and Ideals (1948) don't use the term. (I should point out, for other readers, that Trav is concerned for his joke: Why are fields immoral? Because they have no proper ideals ;-) Paul August ☎ 05:29, 21 September 2005 (UTC)

Ideal Operations

The operations of Ideals are briefly mentioned at the buttom of the article. However, for beginners like me, it is not really clear what does sum and product of ideals mean. I feel it will be good to give the exact definition of them. Here is what I think they mean:

Let I and J be ideals of R

$I+J:=\{a+b|a\in I{\mbox{ and }}b\in J\}$

$IJ:=\{ab|a\in I{\mbox{ and }}b\in J\}$

Please verify this definition.

Also, if this def'n is right, is there any relationship between sum and union? The union of 2 ideals is a subset of the sum of those 2 ideals?

Thanks!-67.43.133.24 03:02, 27 September 2005 (UTC)

You are right about the sum. For the product, you do all those products, but then have to add them up in all possible combinations, as otherwise you don't get an ideal. That's what the phrase "generated by the products" means. So:

$IJ:=\{a_{1}b_{1}+\dots +a_{n}b_{n}|a_{i}\in I{\mbox{ and }}b_{i}\in J,i=1,2,\dots ,n,n=1,2,\dots \}$

And you are right that the union is contained in the sum. This because an element a in I can be written as a+0, and an element b in J can be written as 0+b, so they are both in the sum.

Feel free to add this to the article. Oleg Alexandrov 03:20, 27 September 2005 (UTC)

Thanks for the answer! I already add it to the article. I love wikipedia! -67.43.133.24 04:43, 27 September 2005 (UTC)

Recent reversion

I undid an edit of Zdenes for the following several reasons:

1. An edit summary was missing (one should explain why one changes something)
2. One switched from denoting rings from R to M in the middle of article.
3. One introduced the notation o' which was not explained. Using plain 0 was fine enough I think.
4. I did not see the need for using \forall x,y etc, using plain English "for all" was more acceptable.

I guess some of that may find its way back in the article, but I would like to ask Zdenes why he/she made these changes to start with. Comments? Oleg Alexandrov (talk) 22:57, 29 December 2005 (UTC)

I intended to make that part a bit more accurate. And I thought that denoting the ring R might be too confusing, especially considering the elements 0 and 1 in R. Perhaps it should be emphasized at the beginning that R can be an arbitrary set and is not to be mistaken with the set of real numbers and its elements 0 and 1? Also I think the notation f(1) = 1 and f(a) = 0 is rather problematic. Perhaps it should be emphasized that 0 is the zero element in S and the ones are the identity elements in R, S? and not necessarily the same? But now, having browsed through other wiki articles on similar topic, I can see that using 1 and 0 is the convention here and so my edit was superfluous. Zdenes 11:23, 30 December 2005 (UTC)

Ideals as a Generalisation

The article says that ideals are a generalisation of multiples and divisibility. I know it's kind of related but I think it would be useful to also describe them as a generalisation of the concept of zero. That is the idea of multiplicative absorption in a ring has 0 as it's canonical case and it's generalising it to a set of values - which leads then to a factor ring. --PhiTower 18:02, 12 July 2006 (UTC)

Minimal ideal?

I'm here trying to get a grasp of the notion of a minimal left ideal, but it seems there is only a definition of maximal ideal and left ideal. I'll definitely ask my advisor again for clarification on why he uses this term, but I can't guarantee that I will understand the answer well enough to update this article to include it. --Qrystal (talk) 01:29, 27 January 2010 (UTC)

Definition

Shouldn't the first condition be "for a, b in A, a-b is also in A" rather than "for a, b in A, a+b is also in A"? —Preceding unsigned comment added by 79.180.23.65 (talk) 23:57, 4 May 2010 (UTC)

So rings in wikipedia are assumed to be unital, so the second condition implies that −b is in A for all b in A, hence a−b will be in A. To be clearer, I've made the adjustment though and left this as a footnote. Thanks. RobHar (talk) 03:23, 5 May 2010 (UTC)

The first definition talks of an element x in I, but then proceeds to use 'a' for said element. I've never edited a wiki before, so I'm not sure quite how I'm supposed to sign off. Hopefully someone better versed can enact this change. —Preceding unsigned comment added by 129.215.5.254 (talk) 11:05, 11 May 2010 (UTC)

Ideal generated by a set

The interruptions explaining the equivalence between left, right, and two sided ideals make the second paragraph very unpleasant to read. Is there no way to summarise this equivalence at the end of the paragraph, rather than introducing these ungainly parentheses? —Preceding unsigned comment added by 129.215.5.254 (talk) 11:24, 11 May 2010 (UTC)

Notation

The notation (S) for the ideal generated by S is not explained, although used. Madyno (talk) 19:20, 24 August 2017 (UTC)

"However, in other rings, the ideals may be distinct from the ring elements"

If ideals are subsets of a ring, how can an ideal be distinct from ring elements since an ideal is by definition always a set of ring elements? Acorrector (talk) 10:08, 19 June 2020 (UTC)

The wording was not great. I have reworded it to make more sense. —Quondum 16:45, 3 January 2021 (UTC)

rng/ring confusion

WP seems to be very (unital-)ring centric, and this article is perhaps a good illustration. As far as I can tell, rng theory was developed before rings received as much attention as they receive today, and the concept of an ideal dates to this time and applied usefully in this concept. I imagine that "ring theory" really means "the theory of rngs", but the terminology has drifted. Yet, the article Ring theory uses the term "ring" (which in WP specifically refers to unital rings) and links to Ring (mathematics) with no mention of rngs.

Ideals specifically apply in the broader rng context and were, it seems, largely developed in that context – for example, Emmy Noether gave axioms for defining an ideal in the context of rngs. They are also more "at home" in that context, since they are subrngs but not necessarily subrings. Should we not do one of:

(a) make a clear note that in this article that "ring" refers to "rng" (as no doubt many references intended it), or

(b) change the wording so that each use of "ring" is adjusted to "ring" or "rng", according to its meaning in each reference?

—Quondum 17:10, 3 January 2021 (UTC)

I'm in agreement that many ring-theory articles have tendency to be "(unital-)ring centric". I myself am guilty of imposing that tendency :) In my view, this tendency is a correct one to have since *nowadays" many textbooks on abstract algebra more or less ignore rngs. (Old texts e.g., Jacobson, "structure of rings", on the other hand, have a more complete discussion of rngs theory). It seems that the consensus today is that you need unity to develop a good-behaving "ring theory". You can say this is unfortunate since there are certainly many good examples of rngs. Since Wikipedia articles need to be reflections of a predominant theory (regardless of that theory is good one or not), I think it is correct to focus on ideals in (unital) rings here. -- Taku (talk) 07:16, 4 January 2021 (UTC)

Are you sure? I will not argue that most focus is on unital rings, and that is not a problem. Rewriting history in WP, on the other hand, is a problem. Pose the question: Which subdiscipline of mathematics does the study of rngs (irrespective of whether it is neglected) belong to? I suspect the answer is unambiguously to ring theory. Ring theory has a history that started with the study of what WP now calls rngs, and this should be reflected in our account of ring theory, but I see no hint of it in the article on the subject. Given that Noether's axioms for a (German) "Ring" are what we here call rngs (not rings), what were the objects that were termed "Noetherian rings"? The answer to this in not obvious, and the unwary will assume that they include a 1 by definition (maybe they do by definition, by theorem, by historical redefinition of the term, or maybe they do not necessarily at all), but my point is that I cannot figure out by reading WP what the case is. Ring theorists use rngs all the time as an essential tool (ideals are an example). Anyhow, just because most modern authors might focus on unital rings does not give us, as WP editors, carte blanch to re-interpret the history as though they were always proving theorems only for unital rings because they used that name for what we call rngs now. We need to take care to check their axioms when we relate what they worked on. —Quondum 14:31, 4 January 2021 (UTC)

So everyone here is agreed on the status of identities, we're just asking what's the best way to alert the reader to remain flexible while interpreting this article in the context of all the other articles, right?

I'm in favor of a solid note establishing (a) above, with additional inline notes when identity is required. There's no doubt in my mind the status of identities should be neutral in this article, but I don't think wholesale switchover in (b) is worth it. Rschwieb (talk) 15:39, 4 January 2021 (UTC)

It's worth a try. I expect we'll have links like ring and ring homomorphism. —Quondum 20:16, 4 January 2021 (UTC)

I think that Wikipedia should reflect the actual usage in mathematical books and journals outside Wikipedia. Outside Wikipedia that there are some that deal with (unital) rings and some with rngs, but to act as if they are occurring with equal frequency is not a neutral point of view at all, because the reality is that most of current ring theory (outside of some American undergraduate textbooks) is concerned with the unital case. Of course we don't want to rewrite history; I don't think anyone is advocating that.

As for the Wikipedia article Ring theory, almost all the specific topics and theorems listed there are about unital rings - you can see for yourself. So I don't think it makes sense to say at the top of that page that "ring" means "rng" (anyway, that would go against the Manual of Style). I think it is best to simply to say that "Ring theory" sometimes includes the study of rngs, which I think accurately describes how the term is used in the outside literature; I added a note on the Ring theory article saying this. With this point of view, it is correct to say that Noether developed "ring theory". One could say instead that Noether developed "rng theory", but I don't think people write this way outside of Wikipedia, so I don't think we should start doing so. Ebony Jackson (talk) 21:52, 4 January 2021 (UTC)

You seem to think I'm advocating for something stronger than I am. For example, on the Ring theory page, I would consider it inappropriate, as you do, to change the terminology (or the adopted meaning of the term). No-one has suggested doing so other than on this page (Ideal (ring theory)). I might suggest that the word "sometimes" in your note is overdone, though.

I chose this page as a moderately clear example of where there is no loss of clarity of any statement to frame the statements in terms of rngs because every ring is a rng, but where real encyclopaedic knowledge is being unnecessarily omitted if stated only in terms of rings, and relevant history of the concept includes rngs, whatever we choose to call them. —Quondum 22:46, 4 January 2021 (UTC)

Sorry, Quondum, you are right; I had misunderstood which page(s) you were talking about. Anyway, I think that even in this article Ideal (ring theory) it is tricky to figure out how best to handle things. For example, the definition of principal ideal might at first seem to work in a rng, but is the principal ideal (r) the smallest ideal containing r, or is it rR? These can be different in a rng. Maybe principal ideals are not seriously used in rngs that are not rings? Because of things like this, I would worry about your option (a). On the other hand, option (b) would probably be too unwieldy. Probably figuring out how to handle things will require some research involving reading some advanced books that use "ring" to mean "rng", to see what they do. Ebony Jackson (talk) 23:50, 4 January 2021 (UTC)

The relevant fact here is that Wikipedia is an encyclopedia. If we are writing a textbook, then we get to choose a convention and we might (or might not) like the unity convention for various reasons. But the point Quondum is trying to make is (I think): shouldn't the encyclopedia cover both what is studied and what was studied in the fashion that is today and was in the past? Correct? If so, no, I don't think that's how the encyclopedia articles should be written. Calculus today is the same calculus Newton (and Lebniz) have developed but certainly differs from theirs in the form (e.g., they didn't have the good notion of functions). Wikipedia articles need to be written in the form that many of our target readers expect. The readers educated today would likely expect a ring to have the unity and that's the convention that is needed adopted. We can write the article to cover the rng case simultaneously but that would simply be confusing to readers who are not interested in the rng case. -- Taku (talk) 01:04, 5 January 2021 (UTC)

Ebony brings up a serious obstacle to my suggestion: any generalization from the existing statement would have to be researched, which applies to both (a) and (b). It seems clear that this is a far greater undertaking than I had anticipated, and that my assumption of the generality of the concept may not have survived the modern developments, even in principle.

Taku, close, but not exactly. It would be fine to limit the article to ideals of unital rings. This could be as "an ideal of a ring is ..." instead of "an ideal is ...", for example. I contend that "ring theory" itself does not adequately clarify this. I guess I was reacting to the statement that Emmy Noether contributed significantly the idea (as defined in the article), yet her original definition of an ideal was evidently in the context of rngs (but not stated so here). This gives the impression that the history section might have been written without an understanding that "ring" might have meant "nonunital ring" at the time. If the history section gave a sense of the transition, the reader would not be left wondering about this. —Quondum 01:54, 5 January 2021 (UTC)

Undo the 28 May 2022 Redo of the Introduction/Lead Section

I am reverting to the previous version because:

1. The new version implies the properties of an Ideal under addition are the same as the properties under multiplication, stating that an Ideal is "stable" under both. This implication is wrong. Although both addition and multiplication have the closure property when both operands are elements of the Ideal, only multiplication absorbs an operand from outside the Ideal when the other operand is from the Ideal.
2. The term "stable", used in the redo, is vague. What does it actually mean? The Stability#Mathematics article does not seem to have any meaning that applies here.
3. The last sentence of the first paragraph appears to contradict the first sentence of the next paragraph. In the first paragraph it is the multiples of a given integer (presumably even negative integers) that form an ideal, while in the second paragraph it is the non-negative integers.
4. For someone just starting to learn about Ideals, the previous version is much better than the redo. The previous versions gives the reader a very good idea of what an Ideal is (even if the reader knows nothing of abstract algebra). The redo version only gives them an example (i.e. all the multiples of a given integer), but doesn't tell them what the essential characteristics are.

Obtuse Wombat (talk) 16:39, 27 July 2022 (UTC)