Theorems · Theorem · commutative algebra
isIntegral_of_isIntegral_adjoin_of_mul_eq_one
∀ {R : Type u_1} [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] [inst_2 : Algebra R S] (t s : S),
s * t = 1 → IsIntegral (↥R[s]) t → IsIntegral R tIf t is integral over R[1/t], then it is integral over R.
- Defined in
- Mathlib.RingTheory.Localization.Integral
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites78
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Polynomialproof · cited by 5,681
- Finset.sumproof · cited by 5,195
- Algebra.algebraMapproof · cited by 4,706
- Set.rangeproof · cited by 4,705
- mul_oneproof · cited by 3,885
- AlgHomproof · cited by 3,236
- LE.le.transproof · cited by 3,151
- Nontrivialproof · cited by 2,416
Cited by1
Results whose statement or proof uses this declaration.
- isIntegral_of_isIntegralElem_of_monic_of_natDegree_ltproof · cited by 1