instIsPrincipalIdealRingProd — Mathlib

English

If R and S are principal ideal rings, then R × S is a principal ideal ring.

Русский

Если R и S — п-линейные кольца, то R × S — п-цилиндрическое кольцо (признак principal).

LaTeX

$$instance [IsPrincipalIdealRing R] [IsPrincipalIdealRing S] : IsPrincipalIdealRing (R × S)$$

Lean4

instance [IsPrincipalIdealRing R] [IsPrincipalIdealRing S] : IsPrincipalIdealRing (R × S) where
  principal
    I :=
    by
    rw [I.ideal_prod_eq, ← span_singleton_generator (I.map _), ← span_singleton_generator (I.map (RingHom.snd R S)), ←
      Ideal.span, ← Ideal.span, ← Ideal.span_prod (iff_of_true (by simp) (by simp)), Set.singleton_prod_singleton]
    exact ⟨_, rfl⟩

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