Theorems · Theorem · commutative algebra
Ideal.IsPrime.mul_mem_left_iff
∀ {α : Type u} [inst : Semiring α] {I : Ideal α} [I.IsTwoSided] [I.IsPrime] {x y : α}, x ∉ I → (x * y ∈ I ↔ y ∈ I)- Defined in
- Mathlib.RingTheory.Ideal.Prime
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 20 from the axioms · uses propext, Classical.choice, Quot.sound
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.IsTwoSidedstatement and proof · cited by 179
Cited by4
Results whose statement or proof uses this declaration.
- Ideal.isPrincipal_of_isPrincipal_isLocalizationAway_of_primeproof · cited by 2
- Ideal.exists_not_mem_forall_mem_of_ne_of_liesOverproof · cited by 1
- Algebra.exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOverproof · cited by 1
- Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq_auxproof · cited by 1