Theorems · Inductive type · commutative algebra
IsJacobsonRing
(R : Type u_3) → [CommRing R] → Prop
A ring is a Jacobson ring if for every radical ideal I,
the Jacobson radical of I is equal to I.
See isJacobsonRing_iff_prime_eq and isJacobsonRing_iff_sInf_maximal
for equivalent definitions.
- Defined in
- Mathlib.RingTheory.Jacobson.Ring
- Cited by
- 40 results in Mathlib
- Foundations
- Depth 1 from the axioms · uses no axioms
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites1
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement · cited by 17,173
Cited by43
Results whose statement or proof uses this declaration.
- IsJacobsonRing.outstatement · cited by 6
- isJacobsonRing_iff_prime_eqstatement · cited by 5
- isJacobsonRing_of_finiteTypestatement and proof · cited by 5
- Algebra.QuasiFinite.iff_finite_comap_preimage_singletonproof · cited by 4
- isJacobsonRing_of_surjectivestatement and proof · cited by 4
- finite_of_finite_type_of_isJacobsonRingstatement and proof · cited by 4
- IsLocalization.isMaximal_iff_isMaximal_disjointstatement and proof · cited by 3
- PrimeSpectrum.isJacobsonRing_iff_jacobsonSpacestatement and proof · cited by 2
- Ideal.radical_eq_jacobsonstatement and proof · cited by 2
- AlgebraicGeometry.LocallyOfFiniteType.jacobsonSpaceproof · cited by 2
- PrimeSpectrum.isOpen_singleton_tfae_of_isNoetherian_of_isJacobsonRingstatement and proof · cited by 2
- Polynomial.quotient_mk_comp_C_isIntegral_of_isJacobsonRingstatement and proof · cited by 2