Theorems · Definition · ring theory

GradedModule.isModule

{ιA : Type u_1} →
  {ιM : Type u_2} →
    {A : Type u_4} →
      {M : Type u_5} →
        {σ : Type u_6} →
          {σ' : Type u_7} →
            [inst : AddMonoid ιA] →
              [inst_1 : AddAction ιA ιM] →
                [inst_2 : Semiring A] →
                  (𝓐 : ιA → σ') →
                    [inst_3 : SetLike σ' A] →
                      (𝓜 : ιM → σ) →
                        [inst_4 : AddCommMonoid M] →
                          [inst_5 : Module A M] →
                            [inst_6 : SetLike σ M] →
                              [inst_7 : AddSubmonoidClass σ' A] →
                                [inst_8 : AddSubmonoidClass σ M] →
                                  [SetLike.GradedSMul 𝓐 𝓜] →
                                    [inst_10 : DecidableEq ιA] →
                                      [DecidableEq ιM] → [GradedRing 𝓐] → Module A (DirectSum ιM fun i => ↥(𝓜 i))

The smul multiplication of A on ⨁ i, 𝓜 i from (⨁ i, 𝓐 i) →+ (⨁ i, 𝓜 i) →+ ⨁ i, 𝓜 i turns ⨁ i, 𝓜 i into an A-module

Defined in

Mathlib.Algebra.Module.GradedModule

Cited by

0 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddMonoidAddActionSemiringSetLikeAddCommMonoidModuleSetLikeAddSubmonoidClassAddSubmonoidClassSetLike.GradedSMulDecidableEqDecidableEqGradedRing