Theorems · Theorem · commutative algebra
Ideal.mem_mul_span_singleton
∀ {R : Type u} [inst : Semiring R] {x y : R} {I : Ideal R} [I.IsTwoSided], x ∈ I * Ideal.span {y} ↔ ∃ z ∈ I, z * y = x- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringIdeal.IsTwoSided
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- Idealstatement and proof · cited by 4,748
- Ideal.spanstatement · cited by 948
- Ideal.IsTwoSidedstatement and proof · cited by 179
- Submodule.mem_smul_span_singletonproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- IsGCDMonoid.isPrincipal_of_exists_mul_ne_zero_isPrincipalproof · cited by 1
- Ideal.exists_eq_mul_of_pureproof · cited by 1