Timeline for When can we prove constructively that a ring with unity has a maximal ideal?
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6 events
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| Nov 29, 2009 at 22:57 | comment | added | Greg Kuperberg | I'm a fake expert at best with the algebraic geometry and even worse with the logic. But my impression is that you can make an ordinal height from the Hilbert series of a graded algebra over a field. I referred to the Nullstellensatz above, but the Hilbert series is more directly to the point. | |
| Nov 29, 2009 at 22:45 | comment | added | Pete L. Clark | OK, let's look at the condition on a commutative ring R that there exist an ordinal number (or equivalently, a well-ordered set) O and a map f from the set of nonzero ideals R to O with the property that I strictly contains J implies f(I) < f(J). Such rings are definitely Noetherian. Examples include Euclidean domains and also Samuel's generalization to "ordinally-normed" Euclidean domains. Does the coordinate ring of an affine variety over a field necessarily satisfy this property? | |
| Nov 29, 2009 at 16:09 | comment | added | Greg Kuperberg | Of course it can mean more than one thing. What I mean by it is an order-preserving map from ideals (under reverse inclusion) to ordinals. So that the inductive structure of set of the ideals is constructive. As you argue, it does not really mean that their entire structure is constructive. | |
| Nov 29, 2009 at 7:45 | comment | added | Pete L. Clark | I'm not sure I understand what "constructively Noetherian" means. It seems like it should mean: given a recursive set S of elements of R, give an algorithm to find a finite subset T of R such that <S> = <T>. But I worry that even Z is not constructively Noetherian in this sense. Take for instance the ideal generated by the set x_n = 2^n + e_n, where e_n = 0 if the nth Fourier coefficient of \Delta is nonzero and e_n = 1 otherwise. Then finding a finite generating set for <{x_n}> is equivalent to resolving Lehmer's Conjecture. | |
| Nov 28, 2009 at 19:58 | vote | accept | Qiaochu Yuan | ||
| Nov 28, 2009 at 16:57 | history | answered | Greg Kuperberg | CC BY-SA 2.5 |