Talk:Ideal (ring theory)

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

message: 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)
replies: text:  '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)
thread_title: Is the trivial ideal proper?

message: I undid an edit of Zdenes for the following several reasons:
# An edit summary was missing (one should explain why one changes something)
# One switched from denoting rings from R to M in the middle of article.
# One introduced the notation o' which was not explained. Using plain 0 was fine enough I think.
# 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)
replies: text:  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)
thread_title:  Recent reversion 

message: 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)
thread_title:  Ideals as a Generalisation 

message: 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)
thread_title:  Minimal ideal? 

message: 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)
replies: text: 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)
thread_title:  Definition 

message: 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)
replies: text: The notation (S) for the ideal generated by S is not explained, although used. Madyno (talk) 19:20, 24 August 2017 (UTC)
thread_title:  Ideal generated by a set 

message: 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)
replies: text: The wording was not great.  I have reworded it to make more sense.  —Quondum 16:45, 3 January 2021 (UTC)
thread_title:   "However, in other rings, the ideals may be distinct from the ring elements" 

message: I am reverting to the previous version because:
# 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.
# 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.
# 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.
# 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)
thread_title:  Undo the 28 May 2022 Redo of the Introduction/Lead Section