Theorems · Definition · commutative algebra
Ideal.jacobson
{R : Type u} → [inst : Ring R] → Ideal R → Ideal RThe Jacobson radical of I is the infimum of all maximal (left) ideals containing I.
- Defined in
- Mathlib.RingTheory.Jacobson.Ideal
- Cited by
- 88 results in Mathlib
- Foundations
- Depth 24 from the axioms · uses propext, Quot.sound
- Assumes
- Ring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Ringstatement and proof · cited by 7,463
- Set.ofPredproof · cited by 6,101
- Idealstatement and proof · cited by 4,748
- InfSet.sInfproof · cited by 935
- Ideal.IsMaximalproof · cited by 452
Cited by95
Results whose statement or proof uses this declaration.
- IsLocalRing.maximalIdeal_le_jacobsonstatement · cited by 16
- IsLocalRing.jacobson_eq_maximalIdealstatement · cited by 10
- Submodule.le_of_le_smul_of_le_jacobson_botstatement and proof · cited by 9
- Ideal.le_jacobsonstatement · cited by 9
- IsJacobsonRing.outstatement · cited by 6
- Ideal.jacobson_monostatement and proof · cited by 6
- Submodule.top_ne_ideal_smul_of_le_jacobson_annihilatorstatement and proof · cited by 6
- isJacobsonRing_iff_prime_eqstatement and proof · cited by 5
- isJacobsonRing_of_surjectiveproof · cited by 4
- Submodule.eq_bot_of_eq_ideal_smul_of_le_jacobson_annihilatorstatement and proof · cited by 4
- Ideal.jacobson_botstatement · cited by 4
- IsArtinianRing.isNilpotent_jacobson_botstatement · cited by 4