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Theorems · Definition · ring theory

Algebra.adjoinCommSemiringOfComm

Deprecated since 2026-03-11Use Algebra.isMulCommutative_adjoin instead.

(R : Type uR) →
  {A : Type uA} →
    [inst : CommSemiring R] →
      [inst_1 : Semiring A] →
        [inst_2 : Algebra R A] → {s : Set A} → (∀ a ∈ s, ∀ b ∈ s, a * b = b * a) → CommSemiring ↥(Algebra.adjoin R s)

If all elements of s : Set A commute pairwise, then adjoin R s is a non-unital commutative semiring. See note [reducible non-instances].

Defined in
Mathlib.Algebra.Algebra.Subalgebra.Lattice
Cited by
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Foundations
Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CommSemiringSemiringAlgebra
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Set · cited by 53352SetSemiring · cited by 13802SemiringAlgebra · cited by 11388AlgebraCommSemiring · cited by 10911CommSemiringSubalgebra · cited by 1353SubalgebraAlgebra.adjoin · cited by 535Algebra.adjoinIsMulCommutative · cited by 95IsMulCommutativeAlgebra.isMulCommutative_adjoin · cited by 0Algebra.isMulCommutative_…Algebra.adjoinCommSemiringOfC…CITED BYCITES

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