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[Merged by Bors] - feat(RingTheory/Artinian): IsUnit a iff a ∈ R⁰ for an artinian ring R #21084

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[Merged by Bors] - feat(RingTheory/Artinian): IsUnit a iff a ∈ R⁰ for an artinian ring R #21084

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feat(RingTheory/Artinian): `IsUnit a` iff `a ∈ nonZeroDivisors R` for…
mistarro Jan 26, 2025
aab5314
Use `R⁰` notation and add `IsArtinianRing.isUnit_submonoid_eq`.
mistarro Jan 26, 2025
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20 changes: 20 additions & 0 deletions Mathlib/RingTheory/Artinian/Ring.lean
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Expand Up @@ -91,6 +91,26 @@ theorem isField_of_isReduced_of_isLocalRing [IsReduced R] [IsLocalRing R] : IsFi
(IsArtinianRing.equivPi R).trans (RingEquiv.piUnique _) |>.toMulEquiv.isField
_ (Ideal.Quotient.field _).toIsField

section IsUnit

open nonZeroDivisors

/-- If an element of an artinian ring is not a zero divisor then it is a unit. -/
theorem isUnit_of_mem_nonZeroDivisors {a : R} (ha : a ∈ R⁰) : IsUnit a :=
IsUnit.isUnit_iff_mulLeft_bijective.mpr <|
IsArtinian.bijective_of_injective_endomorphism (LinearMap.mulLeft R a)
fun _ _ ↦ (mul_cancel_left_mem_nonZeroDivisors ha).mp

/-- In an artinian ring, an element is a unit iff it is a non-zero-divisor.
See also `isUnit_iff_mem_nonZeroDivisors_of_finite`.-/
theorem isUnit_iff_mem_nonZeroDivisors {a : R} : IsUnit a ↔ a ∈ R⁰ :=
⟨IsUnit.mem_nonZeroDivisors, isUnit_of_mem_nonZeroDivisors⟩

theorem isUnit_submonoid_eq : IsUnit.submonoid R = R⁰ := by
ext; simp [IsUnit.mem_submonoid_iff, isUnit_iff_mem_nonZeroDivisors]

end IsUnit

section Localization

variable (S : Submonoid R) (L : Type*) [CommSemiring L] [Algebra R L] [IsLocalization S L]
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