Theorems · Theorem · commutative algebra
Ideal.finprod_heightOneSpectrum_factorization
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDedekindDomain R] {I : Ideal R},
I ≠ 0 → ∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R), v.maxPowDividing I = IThe ideal I equals the finprod ∏_v v^(val_v(I)).
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 153 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsDedekindDomain
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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 and proof · cited by 4,748
- IsDedekindDomainstatement and proof · cited by 668
- Irreducibleproof · cited by 496
- IsDedekindDomain.HeightOneSpectrumstatement and proof · cited by 338
- finprodstatement and proof · cited by 257
- Associatesproof · cited by 210
- Associates.mkproof · cited by 137
- Associates.factorsproof · cited by 97
- Associates.countproof · cited by 79
- Irreducible.ne_zeroproof · cited by 45
- associated_iff_eqproof · cited by 28
Cited by6
Results whose statement or proof uses this declaration.
- Ideal.map_algebraMap_eq_finsetProd_powproof · cited by 2
- Ideal.finprod_heightOneSpectrum_pow_multiplicityproof · cited by 1
- IsDedekindDomain.isOpen_of_ne_botproof · cited by 1
- Ideal.finprod_heightOneSpectrum_factorization_coeproof · cited by 1
- Ideal.iInf_maxPowDividing_eqproof · cited by 1
- NumberField.FinitePlace.prod_eq_inv_abs_norm_intproof · cited by 1