Theorems · Theorem · commutative algebra
Algebra.restrictScalars_adjoin
∀ (F : Type u_4) [inst : CommSemiring F] {E : Type u_5} [inst_1 : CommSemiring E] [inst_2 : Algebra F E]
(K : Subalgebra F E) (S : Set E), Subalgebra.restrictScalars F (Algebra.adjoin (↥K) S) = Algebra.adjoin F (↑K ∪ S)- Defined in
- Mathlib.RingTheory.Adjoin.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coestatement and proof · cited by 8,199
- Subalgebrastatement and proof · cited by 1,353
- Algebra.adjoinstatement and proof · cited by 535
- Subalgebra.restrictScalarsstatement and proof · cited by 36
- Algebra.adjoin_union_eq_adjoin_adjoinproof · cited by 8
- Algebra.adjoin_eqproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- IntermediateField.sup_toSubalgebra_of_isAlgebraic_rightproof · cited by 6
- IntermediateField.adjoin_intermediateField_toSubalgebra_of_isAlgebraicproof · cited by 5
- Subalgebra.rank_sup_eq_rank_left_mul_rank_of_freeproof · cited by 2