Theorems · Theorem · commutative algebra
FractionalIdeal.isNoetherian
∀ {R₁ : Type u_3} [inst : CommRing R₁] {K : Type u_4} [inst_1 : Field K] [inst_2 : Algebra R₁ K] [IsFractionRing R₁ K]
[IsDomain R₁] [IsNoetherianRing R₁] (I : FractionalIdeal (nonZeroDivisors R₁) K), IsNoetherian R₁ ↥↑IEvery fractional ideal of a Noetherian integral domain is Noetherian.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Submodulestatement · cited by 7,192
- Idealproof · cited by 4,748
- Algebra.algebraMapproof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- nonZeroDivisorsstatement and proof · cited by 895
- IsFractionRingstatement and proof · cited by 738
- FractionalIdealstatement and proof · cited by 423
- IsNoetherianRingstatement and proof · cited by 268
Cited by1
Results whose statement or proof uses this declaration.
- FractionalIdeal.coe_ideal_mul_invproof · cited by 1