Theorems · Definition · commutative algebra

Submodule.quotientEquivDirectSum

{ι : Type u_1} →
  {R : Type u_2} →
    {M : Type u_3} →
      [inst : CommRing R] →
        [inst_1 : AddCommGroup M] →
          [inst_2 : Module R M] →
            [inst_3 : IsDomain R] →
              [inst_4 : IsPrincipalIdealRing R] →
                [inst_5 : Finite ι] →
                  (F : Type u_4) →
                    [inst_6 : CommRing F] →
                      [inst_7 : Algebra F R] →
                        [inst_8 : Module F M] →
                          [inst_9 : IsScalarTower F R M] →
                            (b : Module.Basis ι R M) →
                              {N : Submodule R M} →
                                (h : Module.finrank R ↥N = Module.finrank R M) →
                                  (M ⧸ N) ≃ₗ[F]
                                    DirectSum ι fun i => R ⧸ Ideal.span {Submodule.smithNormalFormCoeffs b h i}

Decompose M⧸N as a direct sum of cyclic R-modules (quotients by the ideals generated by Smith coefficients of N).

Defined in

Mathlib.LinearAlgebra.FreeModule.Finite.Quotient

Cited by

1 results in Mathlib

Foundations

Depth 130 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleIsDomainIsPrincipalIdealRingFiniteCommRingAlgebraModuleIsScalarTower