Theorems · Definition · commutative algebra
IsArtinianRing.quotNilradicalPowEquivPi
(R : Type u_1) →
[inst : CommRing R] →
[IsArtinianRing R] → (n : ℕ) → (R ⧸ nilradical R ^ n) ≃ₐ[R] (I : MaximalSpectrum R) → R ⧸ I.asIdeal ^ nThe quotient of a commutative Artinian ring by a power of its nilradical is isomorphic to a finite product of local rings, namely the quotients by the powers of the maximal ideals.
- Defined in
- Mathlib.RingTheory.Artinian.Module
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsArtinianRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Idealstatement · cited by 4,748
- HasQuotient.Quotientstatement and proof · cited by 2,301
- iInfproof · cited by 1,690
- AlgEquivstatement · cited by 1,681
- RingEquivproof · cited by 1,147
- AlgEquiv.transproof · cited by 108
- RingEquiv.toEquivproof · cited by 101
- IsArtinianRingstatement and proof · cited by 98
- MaximalSpectrumstatement and proof · cited by 73
- MaximalSpectrum.asIdealstatement and proof · cited by 58
- nilradicalstatement · cited by 41
Cited by2
Results whose statement or proof uses this declaration.
- IsArtinianRing.quotNilradicalPowEquivPi_applystatement and proof · cited by 0
- IsArtinianRing.quotNilradicalPowEquivPi_symm_applystatement · cited by 0