Theorems · Theorem · commutative algebra
Algebra.isAlgebraic_adjoin_of_nonempty
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [NoZeroDivisors R]
{s : Set S}, s.Nonempty → ((Algebra.adjoin R s).IsAlgebraic ↔ ∀ x ∈ s, IsAlgebraic R x)- Defined in
- Mathlib.RingTheory.Algebraic.Integral
- Cited by
- 1 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.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Set.Nonemptystatement and proof · cited by 2,627
- IsDomainproof · cited by 2,196
- NoZeroDivisorsstatement and proof · cited by 545
- Algebra.adjoinstatement and proof · cited by 535
- IsAlgebraicstatement and proof · cited by 163
- Algebra.subset_adjoinproof · cited by 109
- isDomain_iff_noZeroDivisors_and_nontrivialproof · cited by 7
- Subalgebra.IsAlgebraicstatement and proof · cited by 6
- IsAlgebraic.nontrivialproof · cited by 5
Cited by1
Results whose statement or proof uses this declaration.
- Algebra.isAlgebraic_adjoin_singleton_iffproof · cited by 1