Theorems · Theorem · ring theory
bijective_algebraMap_of_linearEquiv
∀ {F : Type u_1} {E : Type u_2} [inst : CommSemiring F] [inst_1 : Semiring E] [inst_2 : Algebra F E] (b : F ≃ₗ[F] E),
Function.Bijective ⇑(algebraMap F E)If E is an F-algebra, there exists an F-linear isomorphism from F to E (namely,
E is a free F-module of rank one), then the algebra map from F to E is bijective.
- Defined in
- Mathlib.Algebra.Algebra.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement · cited by 4,706
- LinearEquivstatement and proof · cited by 3,317
- LinearEquiv.toLinearMapproof · cited by 1,171
- Function.Bijectivestatement · cited by 863
- LinearEquiv.bijectiveproof · cited by 35
- bijective_algebraMap_of_linearMapproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Subalgebra.eq_bot_of_rank_le_oneproof · cited by 3
- bijective_algebraMap_int_of_finite_of_unramifiedproof · cited by 0
- Module.nonempty_algEquiv_iff_finrank_eq_oneproof · cited by 0