iff_injective_ringHom — Mathlib

English

If R is a simple ring, then any ring isomorphic image B ≃ R is also simple; i.e., FormallySmooth is preserved by isomorphisms.

Русский

Если R — простое кольцо, то любое кольцо-изображение через изоморфизм тоже простое; простота сохраняется под изоморфизмами.

LaTeX

$$If f: R ≃+* S and IsSimpleRing R, then IsSimpleRing S.$$

Lean4

theorem iff_injective_ringHom (R : Type u) [NonAssocRing R] [Nontrivial R] :
    IsSimpleRing R ↔ ∀ {S : Type u} [NonAssocSemiring S] [Nontrivial S] (f : R →+* S), Function.Injective f :=
  iff_injective_ringHom_or_subsingleton_codomain R |>.trans <|
    ⟨fun H _ _ _ f => H f |>.resolve_right (by simpa [not_subsingleton_iff_nontrivial]), fun H S _ f =>
      subsingleton_or_nontrivial S |>.recOn Or.inr fun _ => Or.inl <| H f⟩

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