Theorems · Theorem · commutative algebra
Algebra.denominator_def
∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : IsPrincipalIdealRing R] [inst_2 : CommRing S]
[inst_3 : Algebra R S] (x : S),
Algebra.denominator R x = Submodule.IsPrincipal.generator ((Subalgebra.toSubmodule (integralClosure R S)).colon {x})- Defined in
- Mathlib.RingTheory.Algebraic.Denominator
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 138 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Submodulestatement · cited by 7,192
- Subalgebrastatement · cited by 1,353
- OrderEmbeddingstatement · cited by 619
- Subalgebra.toSubmodulestatement · cited by 141
- IsPrincipalIdealRingstatement and proof · cited by 131
- integralClosurestatement · cited by 105
- Submodule.colonstatement · cited by 80
- Submodule.IsPrincipal.generatorstatement · cited by 56
Cited by1
Results whose statement or proof uses this declaration.
- Algebra.denominator_dvd_iffproof · cited by 3