Talk:Ideal (ring theory)

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Mathematics rating: Start Class Mid Priority  Field: Algebra

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

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)

Contents

[edit] 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)

[edit] 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_1b_1+ \dots + a_nb_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)

[edit] 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)

[edit] 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)