Theorems · Theorem · commutative algebra
Ideal.Quotient.eq_zero_iff_mem
∀ {R : Type u} [inst : Ring R] {I : Ideal R} {a : R} [inst_1 : I.IsTwoSided], (Ideal.Quotient.mk I) a = 0 ↔ a ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Quotient.Defs
- Cited by
- 74 results in Mathlib
- Foundations
- Depth 83 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.
Cites8
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.Quotient.mkstatement · cited by 610
- Ideal.IsTwoSidedstatement and proof · cited by 179
- Submodule.Quotient.mk_eq_zeroproof · cited by 35
Cited by74
Results whose statement or proof uses this declaration.
- Ideal.mk_kerproof · cited by 59
- Ideal.map_quotient_selfproof · cited by 12
- Ideal.Quotient.mk_eq_mk_iff_sub_memproof · cited by 8
- Algebra.FormallySmooth.comp_surjectiveproof · cited by 6
- ModP.preVal_mkproof · cited by 5
- Ideal.absNorm_memproof · cited by 5
- Ideal.comap_map_eq_self_of_faithfullyFlatproof · cited by 4
- Ideal.Quotient.maximal_of_isFieldproof · cited by 4
- StandardEtalePair.aeval_X_g_mul_mk_Xproof · cited by 3
- StandardEtalePair.hasMap_Xproof · cited by 3
- not_dvd_differentIdeal_of_isCoprime_of_isSeparableproof · cited by 2
- PrimeSpectrum.mem_image_comap_zeroLocus_sdiffproof · cited by 2