Theorems · Theorem · commutative algebra
Ideal.IsPrime.mul_notMem
∀ {α : Type u} [inst : Semiring α] {I : Ideal α}, I.IsPrime → ∀ {x y : α}, x ∉ I → y ∉ I → x * y ∉ I- Defined in
- Mathlib.RingTheory.Ideal.Prime
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses no axioms
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Idealstatement and proof · cited by 4,748
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.IsPrime.mem_or_memproof · cited by 22
Cited by8
Results whose statement or proof uses this declaration.
- Algebra.IsLocalIso.of_span_range_eq_topproof · cited by 3
- Algebra.IsSmoothAt.exists_notMem_isStandardSmoothproof · cited by 2
- Localization.exists_awayMap_bijective_of_localRingHom_bijectiveproof · cited by 1
- Algebra.IsEtaleAt.exists_isStandardEtaleproof · cited by 1
- Localization.localRingHom_bijective_of_saturated_inf_eq_topproof · cited by 1
- Algebra.IsUnramifiedAt.exists_hasStandardEtaleSurjectionOnproof · cited by 1
- Algebra.IsSmoothAt.exists_isStandardEtale_mvPolynomialproof · cited by 1
- Algebra.ZariskisMainProperty.transproof · cited by 0