Theorems · Theorem · commutative algebra
Ideal.IsPrime.mem_or_mem
∀ {α : Type u} [inst : Semiring α] {I : Ideal α}, I.IsPrime → ∀ {x y : α}, x * y ∈ I → x ∈ I ∨ y ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Prime
- Cited by
- 22 results in Mathlib
- Foundations
- Depth 17 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_mem'proof · cited by 5
Cited by22
Results whose statement or proof uses this declaration.
- IsLocalization.AtPrime.isLocalRingproof · cited by 19
- Ideal.map_isPrime_of_surjectiveproof · cited by 11
- Ideal.IsPrime.mem_of_pow_memproof · cited by 10
- Ideal.IsPrime.mul_mem_iff_mem_or_memproof · cited by 8
- Ideal.IsPrime.mul_notMemproof · cited by 8
- IsLocalization.isPrime_iff_isPrime_disjointproof · cited by 6
- Ideal.IsPrime.mem_or_mem_of_mul_eq_zeroproof · cited by 3
- Ideal.isPrime_ideal_prod_topproof · cited by 2
- Ideal.isPrime_of_isPrime_prod_topproof · cited by 2
- Ideal.IsPrime.isPrimaryproof · cited by 2
- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_memproof · cited by 1
- Ideal.sInf_isPrime_of_isChainproof · cited by 1