Theorems · Theorem · ring theory
Algebra.adjoin_eq_ring_closure
∀ {R : Type uR} {A : Type uA} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Algebra R A] (s : Set A),
(Algebra.adjoin R s).toSubring = Subring.closure (Set.range ⇑(algebraMap R A) ∪ s)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Algebra.algebraMapstatement and proof · cited by 4,706
- Set.rangestatement and proof · cited by 4,705
- Subringstatement · cited by 602
- Algebra.adjoinstatement and proof · cited by 535
- Subring.closurestatement and proof · cited by 78
- Subsemiring.closureproof · cited by 53
Cited by2
Results whose statement or proof uses this declaration.
- IsFractionRing.algHom_fieldRange_eq_of_comp_eq_of_range_eqproof · cited by 1
- Algebra.mem_adjoin_iffproof · cited by 1