Theorems · Theorem · commutative algebra

Ideal.mem_quotient_iff_mem_sup

∀ {R : Type u} [inst : Ring R] {I J : Ideal R} [inst_1 : I.IsTwoSided] {x : R},
  (Ideal.Quotient.mk I) x ∈ Ideal.map (Ideal.Quotient.mk I) J ↔ x ∈ J ⊔ I

See also Ideal.mem_quotient_iff_mem in case I ≤ J.

Defined in: Mathlib.RingTheory.Ideal.Quotient.Operations
Cited by: 3 results in Mathlib
Foundations: Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Ring Ideal.IsTwoSided

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Cites13

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 and proof · cited by 610
Ideal.IsTwoSidedstatement and proof · cited by 179
Ideal.Quotient.mk_surjectiveproof · cited by 134
Ideal.mk_kerproof · cited by 59
Ideal.mem_comapproof · cited by 54
Ideal.comap_map_of_surjectiveproof · cited by 30

Cited by3

Results whose statement or proof uses this declaration.

Ideal.mem_quotient_iff_memproof · cited by 3
Ideal.quotientToQuotientRangePowQuotSucc_injectiveproof · cited by 0
Ideal.quotientToQuotientRangePowQuotSucc_surjectiveproof · cited by 0