Theorems · Theorem · ring theory
Algebra.isMulCommutative_adjoin
∀ (R : Type uR) {A : Type uA} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] {s : Set A},
(∀ x ∈ s, ∀ y ∈ s, x * y = y * x) → IsMulCommutative ↥(Algebra.adjoin R s)If all elements of s : Set A commute pairwise, then adjoin R s is commutative.
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coeproof · cited by 8,199
- Subalgebrastatement · cited by 1,353
- Algebra.adjoinstatement and proof · cited by 535
- IsMulCommutativestatement · cited by 95
- Subalgebra.centralizerproof · cited by 25
- IsMulCommutative.of_setLike_mul_commproof · cited by 16
- Set.centralizer_centralizer_comm_of_commproof · cited by 11
- Algebra.adjoin_le_centralizer_centralizerproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- Algebra.adjoinCommSemiringOfCommproof · cited by 0