Theorems · Theorem · commutative algebra

Ideal.IsHomogeneous.mem_iff

∀ {ι : Type u_1} {σ : Type u_2} {A : Type u_3} [inst : Semiring A] [inst_1 : SetLike σ A]
  [inst_2 : AddSubmonoidClass σ A] (𝒜 : ι → σ) [inst_3 : DecidableEq ι] [inst_4 : AddMonoid ι] [inst_5 : GradedRing 𝒜]
  {I : Ideal A}, Ideal.IsHomogeneous 𝒜 I → ∀ {x : A}, x ∈ I ↔ ∀ (i : ι), ↑(((DirectSum.decompose 𝒜) x) i) ∈ I

Defined in

Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal

Cited by

2 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSetLikeAddSubmonoidClassDecidableEqAddMonoidGradedRing