Theorems · Theorem · commutative algebra
Ideal.exists_eq_mul_of_pure
∀ {R : Type u_1} [inst : CommRing R] {I : Ideal R} [I.Pure] {x : R}, x ∈ I → ∃ y ∈ I, x = x * y[Stacks Tag 04PS](https://stacks.math.columbia.edu/tag/04PS) ((1) => (5))
- Defined in
- Mathlib.RingTheory.Ideal.Pure
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIdeal.Pure
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.
- CommRingstatement and proof · cited by 17,173
- Idealstatement and proof · cited by 4,748
- Ideal.spanproof · cited by 948
- Ideal.subset_spanproof · cited by 86
- Ideal.Purestatement and proof · cited by 7
- Ideal.mem_mul_span_singletonproof · cited by 2
- Ideal.inf_eq_mul_of_pureproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.ker_piRingHom_atPrime_eq_of_pureproof · cited by 1