isPrime_iff_of_isPrincipalIdealRing

English

For a principal ideal ring α and a nonzero ideal P ≠ ⊥, P is prime iff there exists p with P = span{p} and p prime.

Русский

Для кольца с Principal идеалами α и неравноправного P ≠ ⊥, P prime тогда и только тогда, когда существует p: P = span{p} и p prime.

LaTeX

$$$P.IsPrime \\iff \\exists p, Prime p \\land P = \\operatorname{span}\\{p\\}.$$$

Lean4

theorem isPrime_iff_of_isPrincipalIdealRing {P : Ideal α} (hP : P ≠ ⊥) : P.IsPrime ↔ ∃ p, Prime p ∧ P = span { p }
    where
  mp
    h := by
    obtain ⟨p, rfl⟩ := Submodule.IsPrincipal.principal P
    exact ⟨p, (span_singleton_prime (by simp [·] at hP)).mp h, rfl⟩
  mpr := by
    rintro ⟨p, hp, rfl⟩
    rwa [span_singleton_prime (by simp [hp.ne_zero])]

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