Theorems · Theorem · commutative algebra

Ideal.IsHomogeneous.isPrime_iff

∀ {ι : Type u_1} {σ : Type u_2} {A : Type u_3} [inst : CommRing A] [inst_1 : AddCommMonoid ι] [inst_2 : LinearOrder ι]
  [IsOrderedCancelAddMonoid ι] [inst_4 : SetLike σ A] [inst_5 : AddSubmonoidClass σ A] {𝒜 : ι → σ}
  [inst_6 : GradedRing 𝒜] {I : Ideal A},
  Ideal.IsHomogeneous 𝒜 I →
    (I.IsPrime ↔
      I ≠ ⊤ ∧ ∀ {x y : A}, SetLike.IsHomogeneousElem 𝒜 x → SetLike.IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I)

Defined in

Mathlib.RingTheory.GradedAlgebra.Radical

Cited by

0 results in Mathlib

Foundations

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

Assumes

CommRingAddCommMonoidLinearOrderIsOrderedCancelAddMonoidSetLikeAddSubmonoidClassGradedRing