{ι : Type u_1} →
  {S : Type u_3} →
    {R : Type u_4} →
      [inst : DecidableEq ι] →
        [inst_1 : AddMonoid ι] →
          [inst_2 : CommSemiring S] →
            [inst_3 : Semiring R] →
              [inst_4 : Algebra S R] →
                (A : ι → Submodule S R) → [inst_5 : SetLike.GradedMonoid A] → (DirectSum ι fun i => ↥(A i)) →ₐ[S] R