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[Merged by Bors] - feat(RingTheory/Artinian) : prime ideals are maximal in aritinial rings #6309

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[Merged by Bors] - feat(RingTheory/Artinian) : prime ideals are maximal in aritinial rings #6309

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@jjaassoonn jjaassoonn commented Aug 2, 2023 •
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prime ideals in artinian rings are maximal

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@jjaassoonn jjaassoonn added awaiting-CI t-algebra Algebra (groups, rings, fields etc) awaiting-review The author would like community review of the PR labels Aug 2, 2023
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alreadydone commented Aug 11, 2023 •
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I have a short proof using the fact that an artinian ring (e.g. R/p) surjects onto every localization of itself. Since R/p is a domain, it also injects into its field of fractions, so it's isomorphic to a field and therefore itself a field. The branch contains some irrelevant stuff and only MulEquiv.isField is needed. Feel free to extract the relevant part to this PR; if you do so, please also make an IsArtinianRing (R ⧸ I) instance (done).

@alreadydone alreadydone mentioned this pull request Aug 12, 2023
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Thanks 🎉

bors merge

@leanprover-community-mathlib4-bot leanprover-community-mathlib4-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 Aug 15, 2023
bors bot pushed a commit that referenced this pull request Aug 15, 2023
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feat(RingTheory/Artinian) : prime ideals are maximal in aritinial rin…
ca250c5 
…gs (#6309)

prime ideals in artinian rings are maximal
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@bors bors bot changed the title feat(RingTheory/Artinian) : prime ideals are maximal in aritinial rings [Merged by Bors] - feat(RingTheory/Artinian) : prime ideals are maximal in aritinial rings Aug 15, 2023
@bors bors bot closed this Aug 15, 2023
@bors bors bot deleted the zjj/isMaximal_of_isPrime branch August 15, 2023 20:48
semorrison pushed a commit that referenced this pull request Aug 17, 2023
@jjaassoonn @semorrison
feat(RingTheory/Artinian) : prime ideals are maximal in aritinial rin…
6aa209c 
…gs (#6309)

prime ideals in artinian rings are maximal
bors bot pushed a commit that referenced this pull request Aug 18, 2023
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feat: generalize algebraic pullback instances (#6536)
7f52802 
Generalizes RingEquiv.noZeroDivisors and RingEquiv.isDomain to MulEquiv

Adds Function.Injective.isLeft/RightCancelMulZero, MulEquiv.toZeroHomClass, and MulEquiv.isField (the last one is useful for #6309)



Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
bors bot pushed a commit that referenced this pull request Aug 25, 2023
@alreadydone @jjaassoonn
refactor: golf #6309 (#6676)
8bf281f 
Also changes a lemma to instance and adds a new instance.



Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com>
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
@alreadydone alreadydone mentioned this pull request Dec 15, 2023
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