Theorems · Theorem · commutative algebra
Ideal.IsPrime.mul_mem_iff_mem_or_mem
∀ {α : Type u} [inst : Semiring α] {I : Ideal α} [I.IsTwoSided], I.IsPrime → ∀ {x y : α}, x * y ∈ I ↔ x ∈ I ∨ y ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Prime
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses no axioms
- Assumes
- SemiringIdeal.IsTwoSided
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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.IsTwoSidedstatement and proof · cited by 179
- Ideal.mul_mem_leftproof · cited by 107
- Ideal.mul_mem_rightproof · cited by 71
- Ideal.IsPrime.mem_or_memproof · cited by 22
Cited by8
Results whose statement or proof uses this declaration.
- Ideal.IsPrime.mul_leproof · cited by 8
- IsLocalization.isPrime_iff_isPrime_disjointproof · cited by 6
- PrimeSpectrum.zeroLocus_singleton_mulproof · cited by 1
- PowerSeries.eq_of_le_of_X_notMem_of_fg_of_isPrimeproof · cited by 1
- ProjectiveSpectrum.zeroLocus_singleton_mulproof · cited by 1
- PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalksproof · cited by 1
- Ideal.IsPrime.mul_mem_right_iffproof · cited by 0