Theorems · Definition · commutative algebra

Submodule.quotientEquivPiZMod

{ι : Type u_1} →
  {M : Type u_3} →
    [inst : AddCommGroup M] →
      [inst_1 : Finite ι] →
        (N : Submodule ℤ M) →
          (b : Module.Basis ι ℤ M) →
            (h : Module.finrank ℤ ↥N = Module.finrank ℤ M) →
              M ⧸ N ≃+ ((i : ι) → ZMod (Submodule.smithNormalFormCoeffs b h i).natAbs)

Quotients by submodules of full rank of free finite ℤ-modules are isomorphic to a direct product of ZMod.

Defined in

Mathlib.LinearAlgebra.FreeModule.Finite.Quotient

Cited by

2 results in Mathlib

Foundations

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

Assumes

AddCommGroupFinite