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 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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