Science Fair Project Encyclopedia
Talk:Ideal (ring theory)
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
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 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)
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