Theorems · Theorem · commutative algebra
Ideal.mem_jacobson_bot
∀ {R : Type u} [inst : CommRing R] {x : R}, x ∈ ⊥.jacobson ↔ ∀ (y : R), IsUnit (x * y + 1)- Defined in
- Mathlib.RingTheory.Jacobson.Ideal
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
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 · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- one_mulproof · cited by 2,841
- Nat.cast_oneproof · cited by 2,501
- mul_commproof · cited by 2,262
- Nat.cast_zeroproof · cited by 1,870
- IsUnitstatement and proof · cited by 1,602
- sub_eq_zeroproof · cited by 407
- add_mulproof · cited by 363
- mul_right_commproof · cited by 108
- Ideal.jacobsonstatement and proof · cited by 88
Cited by4
Results whose statement or proof uses this declaration.
- IsAdicComplete.le_jacobson_botproof · cited by 1
- Ideal.jacobson_bot_polynomial_le_sInf_map_maximalproof · cited by 1
- IsOpenUnits.of_isAdicproof · cited by 0
- isLocalHom_of_le_jacobson_botproof · cited by 0