Theorems · Inductive type · commutative algebra

HomogeneousSubmodule

{ιA : Type u_1} →
  {ιM : Type u_2} →
    {σA : Type u_3} →
      {σM : Type u_4} →
        {A : Type u_5} →
          {M : Type u_6} →
            [inst : Semiring A] →
              [inst_1 : AddCommMonoid M] →
                [inst_2 : Module A M] →
                  (𝒜 : ιA → σA) →
                    (ℳ : ιM → σM) →
                      [inst_3 : DecidableEq ιA] →
                        [inst_4 : AddMonoid ιA] →
                          [inst_5 : SetLike σA A] →
                            [inst_6 : AddSubmonoidClass σA A] →
                              [GradedRing 𝒜] →
                                [inst_8 : DecidableEq ιM] →
                                  [inst_9 : SetLike σM M] →
                                    [inst_10 : AddSubmonoidClass σM M] →
                                      [DirectSum.Decomposition ℳ] →
                                        [inst_12 : VAdd ιA ιM] → [SetLike.GradedSMul 𝒜 ℳ] → Type u_6

For any Semiring A, we collect the homogeneous submodule of A-modules into a type.

Defined in

Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule

Cited by

12 results in Mathlib

Foundations

Depth 12 from the axioms · uses no axioms

Assumes

SemiringAddCommMonoidModuleDecidableEqAddMonoidSetLikeAddSubmonoidClassGradedRingDecidableEqSetLikeAddSubmonoidClassDirectSum.DecompositionVAddSetLike.GradedSMul