Theorems · Definition · commutative algebra

Submodule.orderIsoOfAlgebraMapSurjective

{R : Type u_1} →
  {S : Type u_2} →
    {M : Type u_3} →
      [inst : CommRing R] →
        [inst_1 : Ring S] →
          [inst_2 : AddCommGroup M] →
            [inst_3 : Algebra R S] →
              [inst_4 : Module R M] →
                [inst_5 : Module S M] →
                  [IsScalarTower R S M] → Function.Surjective ⇑(algebraMap R S) → Submodule S M ≃o Submodule R M

Given a commutative ring R, an R-algebra S and an R-module M with a scalar tower IsScalarTower R S M, if the algebra map from R to S is surjective, then this induces an order isomorphism Submodule S M ≃o Submodule R M.

Defined in

Mathlib.Algebra.Algebra.Tower

Cited by

3 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext, Quot.sound

Assumes

CommRingRingAddCommGroupAlgebraModuleModuleIsScalarTower