prime_C_iff — Mathlib

English

If R is Noetherian, then the polynomial ring R[X] is Noetherian.

Русский

Если R — кольцо Ноттере, то кольцо полиномов R[X] также является Ноттере.

LaTeX

$$$[IsNoetherianRing\, R] \\Rightarrow IsNoetherianRing\\, (R[X])$$$

Lean4

theorem prime_C_iff : Prime (C r) ↔ Prime r :=
  ⟨comap_prime C (evalRingHom (0 : R)) fun _ => eval_C, fun hr =>
    by
    have := hr.1
    rw [← Ideal.span_singleton_prime] at hr ⊢
    · rw [← Set.image_singleton, ← Ideal.map_span]
      apply Ideal.isPrime_map_C_of_isPrime hr
    · intro h; apply (this (C_eq_zero.mp h))
    · assumption⟩

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