Theorems · Definition · linear algebra

Module.Basis.restrictScalars

{ι : Type u_1} →
  (R : Type u_3) →
    {M : Type u_5} →
      {S : Type u_7} →
        [inst : CommRing R] →
          [IsDomain R] →
            [inst_2 : Ring S] →
              [Nontrivial S] →
                [inst_4 : AddCommGroup M] →
                  [inst_5 : Algebra R S] →
                    [inst_6 : Module S M] →
                      [inst_7 : Module R M] →
                        [IsScalarTower R S M] →
                          [Module.IsTorsionFree R S] →
                            (b : Module.Basis ι S M) → Module.Basis ι R ↥(Submodule.span R (Set.range ⇑b))

Let b be an S-basis of M. Let R be a CommRing such that Algebra R S has no zero smul divisors, then the submodule of M spanned by b over R admits b as an R-basis.

Defined in

Mathlib.LinearAlgebra.Basis.Submodule

Cited by

11 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDomainRingNontrivialAddCommGroupAlgebraModuleModuleIsScalarTowerModule.IsTorsionFree