Theorems · Theorem · commutative algebra
IsIntegrallyClosedIn.isIntegral_iff
∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A]
[IsIntegrallyClosedIn R A] {x : A}, IsIntegral R x ↔ ∃ y, (algebraMap R A) y = x- Cited by
- 5 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement · cited by 4,706
- IsIntegralstatement · cited by 427
- IsIntegrallyClosedInstatement and proof · cited by 23
- IsIntegralClosure.isIntegral_iffproof · cited by 19
Cited by5
Results whose statement or proof uses this declaration.
- IsIntegrallyClosed.isIntegral_iffproof · cited by 13
- IsGaloisGroup.of_isFractionRingproof · cited by 5
- IsIntegrallyClosedIn.integralClosure_eq_bot_iffproof · cited by 2
- IsIntegrallyClosedIn.exists_algebraMap_eq_of_isIntegral_powproof · cited by 2
- IsIntegrallyClosedIn.exists_algebraMap_eq_of_pow_mem_subalgebraproof · cited by 1