Theorems · Definition · ring theory

DirectSum.decomposeAlgEquiv

{ι : Type u_1} →
  {R : Type u_2} →
    {A : Type u_3} →
      [inst : DecidableEq ι] →
        [inst_1 : AddMonoid ι] →
          [inst_2 : CommSemiring R] →
            [inst_3 : Semiring A] →
              [inst_4 : Algebra R A] →
                (𝒜 : ι → Submodule R A) → [inst_5 : GradedAlgebra 𝒜] → A ≃ₐ[R] DirectSum ι fun i => ↥(𝒜 i)

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

Defined in

Mathlib.RingTheory.GradedAlgebra.Basic

Cited by

5 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqAddMonoidCommSemiringSemiringAlgebraGradedAlgebra