Theorems · Definition · commutative algebra

Submodule.quotientEquivPiSpan

{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] →
                  (N : Submodule R M) →
                    (b : Module.Basis ι R M) →
                      (h : Module.finrank R ↥N = Module.finrank R M) →
                        (M ⧸ N) ≃ₗ[R] (i : ι) → R ⧸ Ideal.span {Submodule.smithNormalFormCoeffs b h i}

We can write the quotient by a submodule of full rank over a PID as a product of quotients by principal ideals.

Defined in

Mathlib.LinearAlgebra.FreeModule.Finite.Quotient

Cited by

0 results in Mathlib

Foundations

Depth 129 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleIsDomainIsPrincipalIdealRingFinite