Theorems · Theorem · commutative algebra
Ideal.mem_jacobson_iff
∀ {R : Type u} [inst : Ring R] {I : Ideal R} {x : R}, x ∈ I.jacobson ↔ ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I- Defined in
- Mathlib.RingTheory.Jacobson.Ideal
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Ring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites33
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Top.topproof · cited by 9,680
- Ringstatement and proof · cited by 7,463
- Set.ofPredproof · cited by 6,101
- Idealstatement and proof · cited by 4,748
- LE.le.transproof · cited by 3,151
- mul_assocproof · cited by 1,667
- le_transproof · cited by 985
- Ideal.spanproof · cited by 948
- neg_mulproof · cited by 654
- Ideal.IsMaximalproof · cited by 452
- sub_add_cancelproof · cited by 344
- neg_subproof · cited by 272
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.mem_jacobson_botproof · cited by 4
- Ideal.exists_mul_add_sub_mem_of_mem_jacobsonproof · cited by 1