Theorems · Theorem · ring theory

NonUnitalStarAlgebra.isMulCommutative_adjoin

∀ (R : Type u) {A : Type v} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : NonUnitalSemiring A]
  [inst_3 : StarRing A] [inst_4 : Module R A] [inst_5 : IsScalarTower R A A] [inst_6 : SMulCommClass R A A]
  [inst_7 : StarModule R A] {s : Set A},
  (∀ x ∈ s, ∀ y ∈ s, x * y = y * x) →
    (∀ a ∈ s, ∀ b ∈ s, a * star b = star b * a) → IsMulCommutative ↥(NonUnitalStarAlgebra.adjoin R s)

If all elements of s : Set A commute pairwise and with elements of star s, then adjoin R s is commutative.

Defined in

Mathlib.Algebra.Star.NonUnitalSubalgebra

Cited by

1 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringStarRingNonUnitalSemiringStarRingModuleIsScalarTowerSMulCommClassStarModule