Theorems · Theorem · commutative algebra
Ideal.disjoint_powers_iff_notMem_of_isPrime
∀ {R : Type u} [inst : CommSemiring R] {I : Ideal R} [I.IsPrime] (y : R), Disjoint ↑(Submonoid.powers y) ↑I ↔ y ∉ I- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringIdeal.IsPrime
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coestatement · cited by 8,199
- Idealstatement and proof · cited by 4,748
- Submonoidstatement · cited by 3,086
- Disjointstatement · cited by 2,201
- Ideal.IsPrimestatement and proof · cited by 827
- Submonoid.powersstatement · cited by 408
- Ideal.IsPrime.isRadicalproof · cited by 14
- Ideal.disjoint_powers_iff_notMemproof · cited by 2
Cited by10
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.mk_mem_carrierproof · cited by 5
- IsLocalization.isMaximal_iff_isMaximal_disjointproof · cited by 3
- Ideal.isPrincipal_of_isPrincipal_isLocalizationAway_of_primeproof · cited by 2
- Algebra.IsSmoothAt.exists_notMem_isStandardSmoothproof · cited by 2
- isJacobsonRing_localizationproof · cited by 1
- Algebra.exists_etale_isIdempotentElem_forall_liesOver_eqproof · cited by 1
- Algebra.exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOverproof · cited by 1
- UniqueFactorizationMonoid.iff_of_isLocalizationAway_of_primeproof · cited by 1
- IsLocalization.isMaximal_of_isMaximal_disjointproof · cited by 1
- Algebra.IsSmoothAt.exists_isStandardEtale_mvPolynomialproof · cited by 1