Theorems · Theorem · commutative algebra
IsLocalization.Away.exists_isIntegral_mul_of_isIntegral_algebraMap
∀ {R : Type u_5} {S : Type u_6} {Sₘ : Type u_7} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing Sₘ]
[inst_3 : Algebra R S] [inst_4 : Algebra S Sₘ] [inst_5 : Algebra R Sₘ] [IsScalarTower R S Sₘ] {r : S},
IsIntegral R r →
∀ [IsLocalization.Away r Sₘ] {x : S}, IsIntegral R ((algebraMap S Sₘ) x) → ∃ n, IsIntegral R (r ^ n * x)If t is R-integral in S[1/r] where r : S is integral over R,
then r ^ n • t is integral in S for some n.
- Defined in
- Mathlib.RingTheory.Localization.Integral
- Cited by
- 2 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.
Cites30
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- CommSemiringproof · cited by 10,911
- RingHomstatement · cited by 10,189
- Polynomialproof · cited by 5,681
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsScalarTowerstatement and proof · cited by 3,896
- Nontrivialproof · cited by 2,416
- Polynomial.Cproof · cited by 1,598
- map_mulproof · cited by 1,137
- Polynomial.mapproof · cited by 806
Cited by2
Results whose statement or proof uses this declaration.
- IsLocalization.Away.exists_isIntegral_mul_of_isIntegral_mk'proof · cited by 0
- mem_adjoin_map_integralClosure_of_isStandardEtaleproof · cited by 0