Theorems · Theorem · commutative algebra
Subring.isIntegrallyClosedIn_iff
∀ {A : Type u_2} [inst : CommRing A] {C : Type u_5} [inst_1 : SetLike C A] [inst_2 : SubringClass C A] {S : C},
IsIntegrallyClosedIn (↥S) A ↔ ∀ ⦃x : A⦄, IsIntegral (↥S) x → x ∈ S- Cited by
- 3 results in Mathlib
- Foundations
- Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingSetLikeSubringClass
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- SetLikestatement and proof · cited by 1,084
- IsIntegralstatement and proof · cited by 427
- FaithfulSMul.algebraMap_injectiveproof · cited by 198
- Subtype.range_valproof · cited by 41
- IsIntegrallyClosedInstatement · cited by 23
- SubringClassstatement and proof · cited by 20
- isIntegrallyClosedIn_iffproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Subring.exists_le_valuationSubring_of_isIntegrallyClosedInproof · cited by 1
- LocalSubring.exists_le_valuationSubring_of_isIntegrallyClosedInproof · cited by 1
- Subring.integralClosure_le_iffproof · cited by 1