Theorems · Theorem · commutative algebra
Ideal.jacobson_eq_iff_jacobson_quotient_eq_bot
∀ {R : Type u} [inst : CommRing R] {I : Ideal R}, I.jacobson = I ↔ ⊥.jacobson = ⊥An ideal I of R is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring R/I is the zero ideal
- Defined in
- Mathlib.RingTheory.Jacobson.Ideal
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 85 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.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- HasQuotient.Quotientstatement and proof · cited by 2,301
- Ideal.mapproof · cited by 692
- Ideal.Quotient.mkproof · cited by 610
- Ideal.comapproof · cited by 443
- le_of_eqproof · cited by 366
- Ideal.jacobsonstatement and proof · cited by 88
- Ideal.mk_kerproof · cited by 59
- RingHom.ker_eq_comap_botproof · cited by 23
Cited by2
Results whose statement or proof uses this declaration.
- isJacobsonRing_of_isIntegralproof · cited by 1
- Ideal.jacobson_bot_polynomial_le_sInf_map_maximalproof · cited by 1