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[Merged by Bors] - feat(RingTheory/Artinian): integral non-zero-divisors are units over artinian rings #21199

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[Merged by Bors] - feat(RingTheory/Artinian): integral non-zero-divisors are units over artinian rings #21199

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In an algebra over an artinian ring, integral elements are either units or zero divisors.
As units are never zero divisors, the fact follows from implication that integral non-zero-divisors are units.

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@github-actions github-actions bot added the large-import Automatically added label for PRs with a significant increase in transitive imports label Jan 28, 2025
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github-actions bot commented Jan 28, 2025 •
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PR summary fc42575eac

Import changes for modified files

No significant changes to the import graph

Import changes for all files
Files Import difference
Mathlib.RingTheory.Artinian.Algebra (new file) 1259

Declarations diff

+ isUnit_iff_nonZeroDivisor_of_isIntegral
+ isUnit_iff_nonZeroDivisor_of_isIntegral'
+ isUnit_of_isIntegral_of_nonZeroDivisor
+ isUnit_of_nonZeroDivisor_of_isIntegral'
+ isUnit_submonoid_eq_of_isIntegral

You can run this locally as follows 
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary
  • The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
  • The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

@github-actions github-actions bot added the t-algebra Algebra (groups, rings, fields, etc) label Jan 28, 2025
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I originally needed this results for algebras over fields but turned out to be a bit more general.
I'm not sure if this is the best place for them, but at least they are next to other similar results.

@github-actions github-actions bot removed the large-import Automatically added label for PRs with a significant increase in transitive imports label Jan 29, 2025
@mistarro mistarro force-pushed the mistarro/isunit-of-nonzerodivisor-of-integral-over-artinian branch 3 times, most recently from 13c4767 to df6392d Compare January 30, 2025 21:44
@jcommelin jcommelin added the awaiting-author A reviewer has asked the author a question or requested changes label Jan 31, 2025
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bors d+

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mathlib-bors bot commented Jan 31, 2025

✌️ mistarro can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

@leanprover-community-mathlib4-bot leanprover-community-mathlib4-bot added delegated and removed awaiting-author A reviewer has asked the author a question or requested changes labels Jan 31, 2025
@mistarro mistarro force-pushed the mistarro/isunit-of-nonzerodivisor-of-integral-over-artinian branch from df6392d to e8482c8 Compare January 31, 2025 18:03
@mistarro mistarro force-pushed the mistarro/isunit-of-nonzerodivisor-of-integral-over-artinian branch from e8482c8 to fc42575 Compare January 31, 2025 18:04
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bors r+

mathlib-bors bot pushed a commit that referenced this pull request Jan 31, 2025
@mistarro
feat(RingTheory/Artinian): integral non-zero-divisors are units over …
7aafb14 
…artinian rings (#21199)

In an algebra over an artinian ring, integral elements are either units or zero divisors.
As units are never zero divisors, the fact follows from implication that integral non-zero-divisors are units.
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mathlib-bors bot commented Jan 31, 2025

Pull request successfully merged into master.

Build succeeded:

@mathlib-bors mathlib-bors bot changed the title feat(RingTheory/Artinian): integral non-zero-divisors are units over artinian rings [Merged by Bors] - feat(RingTheory/Artinian): integral non-zero-divisors are units over artinian rings Jan 31, 2025
@mathlib-bors mathlib-bors bot closed this Jan 31, 2025
@mathlib-bors mathlib-bors bot deleted the mistarro/isunit-of-nonzerodivisor-of-integral-over-artinian branch January 31, 2025 19:29
Julian added a commit that referenced this pull request Feb 2, 2025
@Julian
Merge branch 'factorial-dvd-int' into bhargava-factorial
4aa075f 
* factorial-dvd-int: (143 commits)
  Apply suggestions from code review
  feat(Factorial): k! divides the product of any k successive integers
  feat(CategoryTheory): creation of finite limits (#21320)
  chore: update Mathlib dependencies 2025-02-01 (#21328)
  chore(GroupTheory/SpecificGroups/Alternating.lean): follow last minute review of JX (#21314)
  feat: `‖x‖ₑ.toNNReal = ‖x‖₊` (#21306)
  chore: cleanup imports in Archive/IfNormalization (#21318)
  doc: fix several typos (#21315)
  feat(CategoryTheory): transfer being iso along an iso in the arrow category (#21310)
  chore: delete declarations deprecated between 2024-01 and 2024-07 (#21271)
  feat(Analysis/Normed/Module/Dual): polar in a normed space as a submodule (#20084)
  chore(Data/ZMod/Basic): split `ZMod.valMinAbs` off (#21308)
  feat(GroupTheory/Perm/Centralizer): study the centralizer of a permutation (#17522)
  feat(RingTheory/LocalRing): `IsLocalRing` for subrings (#21168)
  chore: update Mathlib dependencies 2025-02-01 (#21312)
  chore: update Mathlib dependencies 2025-01-31 (#21311)
  feat: generalize `mem_dite` to `Membership α β` (#21262)
  feat: Lemmas for some monomial orders (#16177)
  feat(CategoryTheory): the localized category is monoidal (#12728)
  feat: add function log⁺ (=positive part of the logarithm) and prove standard estimates (#21289)
  feat(RingTheory/WittVector): ring of Witt vectors is p-adically complete (#21295)
  feat(GroupTheory/GroupAction/Blocks): more on blocks (#21284)
  fix(FieldTheory/KrullTopology): make `krullTopology_discreteTopology_of_finiteDimensional` universe polymorphic (#21299)
  feat(RingTheory/Artinian): integral non-zero-divisors are units over artinian rings (#21199)
  refactor(Topology/Gluing): simplify definition of `TopCat.GlueData.Rel` (#20653)
  feat(RingTheory/PowerSeries): binomial series (#20192)
  chore(Mathlib/RingTheory/MvPolynomial): rename MonomiaOrder.lCoeff to MonomialOrder.leadingCoeff  (#21290)
  chore (RingTheory/HahnSeries): fix names that use coeff (#21279)
  feat: let `notation3` distinguish `Prop` vs `Type _ ` vs `Sort _` (#21233)
  chore(MeasureTheory/Function/StronglyMeasurable): split Basic into Basic and AEStronglyMeasurable (#21273)
  feat(CategoryTheory): the monoidal category structure on a localization (#20951)
  feat(Analysis/Complex/Hadamard): generalize Hadamard's three lines theorem (#15009)
  feat(Order/CompleteBooleanAlgebra): Himp in terms of sSup (#20328)
  feat(ENNReal/Basic): add `ofNat_ne_top` and `top_ne_ofNat` (#14486)
  feat: Function.const as a PartialEquiv (#21137)
  chore(NonZeroDivisors): don't import rings (#20871)
  feat(Data/Set/Lattice): insert distributivity with iUnion/iInter (#21267)
  feat(GroupTheory/SpecificGroups/AlternatingGroup): subgroups of index 2 of Equiv.Perm (#21190)
  feat(GroupTheory/GroupAction/Transitive): basic results on transitive actions (#21285)
  perf(MeasureTheory/Function/LpSpace.lean): speed up (#21179)
  feat(Order): order isomorphisms from `Fin n` for small `n` (#21120)
  refactor(Topology/Group): turn morphisms in ProfiniteGrp into one field structures (#20740)
  feat: Sylow's first theorem for elementary `p`-groups (#21072)
  chore(Submonoid/Membership): don't import `MonoidWithZero` (#20748)
  refactor(Algebra/Algebra/Pi): cleanup and renaming (#21213)
  feat(GroupTheory/IndexNormal): subgroups of small index are normal (#21186)
  feat(Algebra/Group/Action): add definition of equidecomposition (#16936)
  feat(CategoryTheory/Subpresheaf): equalizer (#21096)
  feat: add lemmas about products of `Matrix.stdBasisMatrix` (#21204)
  chore: update Mathlib dependencies 2025-01-31 (#21282)
  ...
@alreadydone alreadydone mentioned this pull request Feb 2, 2025
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jt496 pushed a commit that referenced this pull request Feb 3, 2025
@mistarro @jt496
feat(RingTheory/Artinian): integral non-zero-divisors are units over …
b6c347c 
…artinian rings (#21199)

In an algebra over an artinian ring, integral elements are either units or zero divisors.
As units are never zero divisors, the fact follows from implication that integral non-zero-divisors are units.
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