isAlgebraic_ringHom_iff_of_comp_eq

English

For f, g related by a composition equality, IsAlgebraic S (g a) iff IsAlgebraic R a, when g is a ring equivalence-like map.

Русский

Для f, g связанные равенством композиции, IsAlgebraic(S, g(a)) эквивално IsAlgebraic(R, a) при эквививном виде g.

LaTeX

$$$IsAlg(S, g(a)) \iff IsAlg(R, a)$ under composition equality and equivariant assumptions.$$

Lean4

theorem isAlgebraic_ringHom_iff_of_comp_eq (hg : Function.Injective g)
    (h : RingHom.comp (algebraMap S B) f = RingHom.comp g (algebraMap R A)) {a : A} :
    IsAlgebraic S (g a) ↔ IsAlgebraic R a :=
  ⟨fun H ↦ H.of_ringHom_of_comp_eq f g (EquivLike.surjective f) hg h, fun H ↦
    H.ringHom_of_comp_eq f g (EquivLike.injective f) h⟩

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