Theorems · Definition · group theory

DirectSum.decomposeLinearEquiv

{ι : Type u_1} →
  {R : Type u_2} →
    {M : Type u_3} →
      [inst : DecidableEq ι] →
        [inst_1 : Semiring R] →
          [inst_2 : AddCommMonoid M] →
            [inst_3 : Module R M] →
              (ℳ : ι → Submodule R M) → [DirectSum.Decomposition ℳ] → M ≃ₗ[R] DirectSum ι fun i => ↥(ℳ i)

If M is graded by ι with degree i component ℳ i, then it is isomorphic as a module to a direct sum of components.

Defined in

Mathlib.Algebra.DirectSum.Decomposition

Cited by

15 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqSemiringAddCommMonoidModuleDirectSum.Decomposition