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[Merged by Bors] - feat(RingTheory/Localization): add facts about localization at minimal prime ideals #11201

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[Merged by Bors] - feat(RingTheory/Localization): add facts about localization at minimal prime ideals #11201

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uniwuni
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@uniwuni uniwuni commented Mar 6, 2024 •
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Show that localization at minimal primes results in rings with only a single prime ideal, implying that every non-unit element is nilpotent.

Co-authored-by: Junyan Xu junyanxumath@gmail.com

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@uniwuni uniwuni added awaiting-review The author would like community review of the PR t-algebra Algebra (groups, rings, fields etc) labels Mar 6, 2024
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uniwuni commented Mar 6, 2024

It wouldn't take much effort to show that the prime spectrum of these rings is a single point but I'm not sure where this would fit exactly

faenuccio
faenuccio reviewed Mar 11, 2024
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It wouldn't take much effort to show that the prime spectrum of these rings is a single point but I'm not sure where this would fit exactly

I agree that this should be added, probably in Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic (where, for instances, there are results like PrimeSpectrum.t1Space_iff_isField )

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uniwuni commented Mar 12, 2024

I agree that this should be added, probably in Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic (where, for instances, there are results like PrimeSpectrum.t1Space_iff_isField )

Done. I think the Uniqueness result should be enough for the moment

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alreadydone commented Mar 13, 2024 •
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Here is a slight refactor that introduces IsLocalization.AtPrime.orderIsoOfPrime and IsLocalization.AtPrime.prime_unique_of_minimal. Feel free to merge if it looks good.

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Apart from @alreadydone 's comment, the whole PR now LGTM.

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uniwuni commented Mar 13, 2024

Here is a slight refactor that introduces IsLocalization.AtPrime.orderIsoOfPrime and IsLocalization.AtPrime.prime_unique_of_minimal. Feel free to merge if it looks good.

If you add docs to the def to make it compile, I'll do so, sure!

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I've now added a docstring, but this is something you could have done :)

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Merge pull request #11361 from leanprover-community/jyxu/localization…
65c14a1 
…_minimalPrime

Jyxu/localization minimal prime
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LGTM

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Thanks 🎉
maintainer merge

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🚀 Pull request has been placed on the maintainer queue by alreadydone.

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Thanks!

bors merge

@github-actions github-actions bot added ready-to-merge This PR has been sent to bors. and removed awaiting-review The author would like community review of the PR labels Mar 14, 2024
mathlib-bors bot pushed a commit that referenced this pull request Mar 14, 2024
@uniwuni @alreadydone
feat(RingTheory/Localization): add facts about localization at minima…
415f214 
…l prime ideals (#11201)

Show that localization at minimal primes results in rings with only a single prime ideal, implying that every non-unit element is nilpotent.

Co-authored-by: Junyan Xu <junyanxumath@gmail.com>



Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Co-authored-by: uniwuni <95649083+uniwuni@users.noreply.github.com>
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mathlib-bors bot commented Mar 14, 2024

Pull request successfully merged into master.

Build succeeded:

@mathlib-bors mathlib-bors bot changed the title feat(RingTheory/Localization): add facts about localization at minimal prime ideals [Merged by Bors] - feat(RingTheory/Localization): add facts about localization at minimal prime ideals Mar 14, 2024
@mathlib-bors mathlib-bors bot closed this Mar 14, 2024
@mathlib-bors mathlib-bors bot deleted the uniwuni/localization_min_ideals branch March 14, 2024 19:01
dagurtomas pushed a commit that referenced this pull request Mar 22, 2024
@uniwuni @alreadydone @dagurtomas
feat(RingTheory/Localization): add facts about localization at minima…
63ac9f5 
…l prime ideals (#11201)

Show that localization at minimal primes results in rings with only a single prime ideal, implying that every non-unit element is nilpotent.

Co-authored-by: Junyan Xu <junyanxumath@gmail.com>



Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Co-authored-by: uniwuni <95649083+uniwuni@users.noreply.github.com>
utensil pushed a commit that referenced this pull request Mar 26, 2024
@uniwuni @alreadydone @utensil
feat(RingTheory/Localization): add facts about localization at minima…
78f36d4 
…l prime ideals (#11201)

Show that localization at minimal primes results in rings with only a single prime ideal, implying that every non-unit element is nilpotent.

Co-authored-by: Junyan Xu <junyanxumath@gmail.com>



Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Co-authored-by: uniwuni <95649083+uniwuni@users.noreply.github.com>
Louddy pushed a commit that referenced this pull request Apr 15, 2024
@uniwuni @alreadydone @Louddy
feat(RingTheory/Localization): add facts about localization at minima…
2bd54a3 
…l prime ideals (#11201)

Show that localization at minimal primes results in rings with only a single prime ideal, implying that every non-unit element is nilpotent.

Co-authored-by: Junyan Xu <junyanxumath@gmail.com>



Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Co-authored-by: uniwuni <95649083+uniwuni@users.noreply.github.com>
uniwuni added a commit that referenced this pull request Apr 19, 2024
@uniwuni @alreadydone
feat(RingTheory/Localization): add facts about localization at minima…
3c94642 
…l prime ideals (#11201)

Show that localization at minimal primes results in rings with only a single prime ideal, implying that every non-unit element is nilpotent.

Co-authored-by: Junyan Xu <junyanxumath@gmail.com>



Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Co-authored-by: uniwuni <95649083+uniwuni@users.noreply.github.com>
callesonne pushed a commit that referenced this pull request Apr 22, 2024
@uniwuni @alreadydone @callesonne
feat(RingTheory/Localization): add facts about localization at minima…
2e26618 
…l prime ideals (#11201)

Show that localization at minimal primes results in rings with only a single prime ideal, implying that every non-unit element is nilpotent.

Co-authored-by: Junyan Xu <junyanxumath@gmail.com>



Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Co-authored-by: uniwuni <95649083+uniwuni@users.noreply.github.com>
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