Theorems · Theorem · commutative algebra
Subring.eq_iInf_of_isIntegrallyClosedIn
∀ {K : Type u_3} [inst : Field K] {R : Subring K} [IsIntegrallyClosedIn (↥R) K], R = ⨅ V, (↑V).toSubringA subring integrally closed in a field is the intersection of valuation subrings containing it.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 143 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldIsIntegrallyClosedIn
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Fieldstatement and proof · cited by 7,404
- le_antisymmproof · cited by 2,068
- iInfstatement and proof · cited by 1,690
- le_rflproof · cited by 1,558
- Subringstatement and proof · cited by 602
- ValuationSubringstatement and proof · cited by 187
- le_iInfproof · cited by 102
- iInf_le_of_leproof · cited by 62
- of_not_notproof · cited by 51
- ValuationSubring.toSubringstatement and proof · cited by 32
- IsIntegrallyClosedInstatement and proof · cited by 23
- Subring.exists_le_valuationSubring_of_isIntegrallyClosedInproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- iInf_valuationSubring_supersetproof · cited by 0