Theorems · Theorem · commutative algebra
Algebra.denominator_dvd_iff
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : IsPrincipalIdealRing R] [inst_2 : CommRing S]
[inst_3 : Algebra R S] {r : R} {x : S}, Algebra.denominator R x ∣ r ↔ IsIntegral R (r • x)- Defined in
- Mathlib.RingTheory.Algebraic.Denominator
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 139 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- IsIntegralstatement and proof · cited by 427
- Subalgebra.toSubmoduleproof · cited by 141
- IsPrincipalIdealRingstatement and proof · cited by 131
- integralClosureproof · cited by 105
- Submodule.colonproof · cited by 80
- Submodule.IsPrincipal.mem_iff_generator_dvdproof · cited by 11
- Submodule.mem_colon_singletonproof · cited by 8
- Algebra.denominatorstatement · cited by 5
- Subalgebra.mem_toSubmoduleproof · cited by 5
Cited by3
Results whose statement or proof uses this declaration.
- Algebra.natDenominator_dvd_iffproof · cited by 1
- IsAlgebraic.denominator_ne_zeroproof · cited by 1
- Algebra.isIntegral_denominator_smulproof · cited by 0