Theorems · Theorem · ring theory
AlgHom.range_eq_top
∀ {R : Type u} {A : Type v} {B : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A]
[inst_3 : Semiring B] [inst_4 : Algebra R B] (f : A →ₐ[R] B), f.range = ⊤ ↔ Function.Surjective ⇑f- Cited by
- 12 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- DFunLike.coestatement · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Top.topstatement · cited by 9,680
- AlgHomstatement and proof · cited by 3,236
- Subalgebrastatement · cited by 1,353
- AlgHom.rangestatement · cited by 169
- Algebra.eq_top_iffproof · cited by 13
Cited by12
Results whose statement or proof uses this declaration.
- Algebra.FiniteType.of_finiteType_tensorProduct_of_faithfullyFlatproof · cited by 3
- IsLocalRing.adjoin_residue_eq_top_iff_adjoin_eq_topproof · cited by 2
- Algebra.FinitePresentation.of_restrict_scalars_finitePresentationproof · cited by 2
- Algebra.adjoin_root_eq_top_of_isSplittingFieldproof · cited by 1
- Algebra.not_isStronglyTranscendental_of_weaklyQuasiFiniteAtproof · cited by 1
- IsAdjoinRoot.adjoin_root_eq_topproof · cited by 1
- RingCon.range_mkₐproof · cited by 0
- AdjoinRoot.Minpoly.toAdjoin.surjectiveproof · cited by 0
- Subalgebra.map_comap_eq_self_of_surjectiveproof · cited by 0
- Polynomial.IsSplittingField.of_algEquivproof · cited by 0
- DividedPowerAlgebra.lift_surjectiveproof · cited by 0