instCommSemiring — Mathlib

English

If A is a commutative semiring and S is a submonoid of R with localization, LocalizedModule S A becomes a commutative semiring, where multiplication is commutative on the numerators and denominators accordingly.

Русский

Если A — коммутативное полугрей, и S задаёт локализацию, то LocalizedModule S A образует коммутативное полугрупповое кольцо: умножение коммутативно по числителям и знаменателям.

LaTeX

$$$\\mathrm{LocalizedModule}(S,A)$ is a CommSemiring$$

Lean4

instance {A : Type*} [CommSemiring A] [Algebra R A] {S : Submonoid R} : CommSemiring (LocalizedModule S A) :=
  { show Semiring (LocalizedModule S A) by infer_instance with
    mul_comm := by
      rintro ⟨a₁, s₁⟩ ⟨a₂, s₂⟩
      exact mk_eq.mpr ⟨1, by simp only [one_smul, mul_comm]⟩ }

Похожие утверждения