Theorems · Theorem · commutative algebra

Algebra.adjoin_eq_adjoin_union

∀ (R : Type uR) {A : Type uA} {B : Type uB} [inst : CommSemiring R] [inst_1 : CommSemiring A] [inst_2 : Algebra R A]
  [inst_3 : CommSemiring B] [inst_4 : Algebra R B] [inst_5 : Algebra A B] [inst_6 : IsScalarTower R A B] (s : Set A)
  (t : Set B),
  Algebra.adjoin R s = ⊤ →
    Subalgebra.restrictScalars R (Algebra.adjoin A t) = Algebra.adjoin R (⇑(algebraMap A B) '' s ∪ t)

If A is spanned over R by s, then the algebra spanned over A by t is the equal to the algebra spanned over R by s ∪ t.

Defined in

Mathlib.RingTheory.Adjoin.Basic

Cited by

1 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraCommSemiringAlgebraAlgebraIsScalarTower