Theorems · Theorem · commutative algebra

Submodule.restrictScalars_span

∀ (R : Type u) (A : Type w) {M : Type v₁} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A]
  [inst_3 : AddCommMonoid M] [inst_4 : Module R M] [inst_5 : Module A M] [inst_6 : IsScalarTower R A M],
  Function.Surjective ⇑(algebraMap R A) →
    ∀ (X : Set M), Submodule.restrictScalars R (Submodule.span A X) = Submodule.span R X

If A is an R-algebra such that the induced morphism R →+* A is surjective, then the R-module generated by a set X equals the A-module generated by X.

Defined in

Mathlib.Algebra.Algebra.Tower

Cited by

12 results in Mathlib

Foundations

Depth 26 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringSemiringAlgebraAddCommMonoidModuleModuleIsScalarTower