Theorems · Definition · ring theory
Module.jacobson
(R : Type u_1) → (M : Type u_3) → [inst : Ring R] → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → Submodule R M
The Jacobson radical of an R-module M is the infimum of all maximal submodules in M.
- Defined in
- Mathlib.RingTheory.Jacobson.Radical
- Cited by
- 20 results in Mathlib
- Foundations
- Depth 25 from the axioms · uses propext, Quot.sound
- Assumes
- RingAddCommGroupModule
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.
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- Set.ofPredproof · cited by 6,101
- InfSet.sInfproof · cited by 935
- IsCoatomproof · cited by 114
Cited by21
Results whose statement or proof uses this declaration.
- Ring.jacobsonproof · cited by 37
- Module.map_jacobson_of_ker_lestatement and proof · cited by 6
- Module.le_comap_jacobsonstatement and proof · cited by 6
- IsSemisimpleModule.jacobson_eq_botstatement and proof · cited by 4
- Module.map_jacobson_lestatement · cited by 2
- Module.jacobson_lt_topstatement · cited by 2
- Module.jacobson_eq_bot_of_injectivestatement and proof · cited by 1
- Module.jacobson_le_of_eq_botstatement and proof · cited by 1
- Module.jacobson_pi_eq_botstatement and proof · cited by 1
- Module.jacobson_pi_lestatement and proof · cited by 1
- Module.jacobson_quotient_of_lestatement and proof · cited by 1
- Ring.coe_jacobson_quotientstatement and proof · cited by 1