Theorems · Theorem · ring theory
Algebra.adjoin_eq_span
∀ (R : Type uR) {A : Type uA} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (s : Set A),
Subalgebra.toSubmodule (Algebra.adjoin R s) = Submodule.span R ↑(Submonoid.closure s)- Cited by
- 13 results in Mathlib
- Foundations
- Depth 73 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.
Cites33
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- 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.coestatement and proof · cited by 8,199
- Submodulestatement · cited by 7,192
- Algebra.algebraMapproof · cited by 4,706
- Set.rangeproof · cited by 4,705
- Submonoidstatement · cited by 3,086
- le_antisymmproof · cited by 2,068
- mul_assocproof · cited by 1,667
Cited by13
Results whose statement or proof uses this declaration.
- Algebra.adjoin_union_coe_submoduleproof · cited by 3
- AddMonoidAlgebra.finiteType_iff_fgproof · cited by 3
- IsLocalization.finiteType_of_monoid_fgproof · cited by 2
- Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoinproof · cited by 2
- exists_subalgebra_of_fgproof · cited by 1
- Submonoid.adjoin_eq_span_of_eq_spanproof · cited by 1
- HomogeneousLocalization.Away.span_mk_prod_pow_eq_topproof · cited by 1
- IsAlmostIntegral.isIntegral_of_nonZeroDivisors_le_comapproof · cited by 1
- multiple_mem_adjoin_of_mem_localization_adjoinproof · cited by 1
- Algebra.adjoin_toSubmodule_leproof · cited by 1
- StarAlgebra.adjoin_eq_spanproof · cited by 1
- Algebra.adjoin_nonUnitalSubalgebra_eq_spanproof · cited by 0