Theorems · Theorem · commutative algebra
Ideal.mem_quotient_iff_mem
∀ {R : Type u} [inst : Ring R] {I J : Ideal R} [inst_1 : I.IsTwoSided],
I ≤ J → ∀ {x : R}, (Ideal.Quotient.mk I) x ∈ Ideal.map (Ideal.Quotient.mk I) J ↔ x ∈ JSee also Ideal.mem_quotient_iff_mem_sup if the assumption I ≤ J is not available.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RingIdeal.IsTwoSided
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Idealstatement and proof · cited by 4,748
- HasQuotient.Quotientstatement · cited by 2,301
- Ideal.mapstatement · cited by 692
- Ideal.Quotient.mkstatement · cited by 610
- Ideal.IsTwoSidedstatement and proof · cited by 179
- sup_eq_leftproof · cited by 71
- Ideal.mem_quotient_iff_mem_supproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- Algebra.FormallyUnramified.lift_uniqueproof · cited by 3
- Ideal.height_le_height_add_of_liesOverproof · cited by 1
- Ideal.exists_coeff_mem_comap_sdiff_comap_of_root_mem_sdiffproof · cited by 1