Theorems · Theorem · commutative algebra
Ideal.mem_map_iff_of_surjective
∀ {R : Type u} {S : Type v} {F : Type u_1} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : FunLike F R S] (f : F)
[RingHomClass F R S], Function.Surjective ⇑f → ∀ {I : Ideal R} {y : S}, y ∈ Ideal.map f I ↔ ∃ x ∈ I, f x = y- Defined in
- Mathlib.RingTheory.Ideal.Maps
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Quot.sound
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 and proof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- SetLike.coeproof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- FunLikestatement and proof · cited by 2,560
- Ideal.mapstatement and proof · cited by 692
- RingHomClassstatement and proof · cited by 193
- Set.mem_imageproof · cited by 131
- Ideal.mem_map_of_memproof · cited by 71
- Ideal.mem_image_of_mem_map_of_surjectiveproof · cited by 6
Cited by19
Results whose statement or proof uses this declaration.
- Ideal.map_isPrime_of_surjectiveproof · cited by 11
- Algebra.Generators.Cotangent.exactproof · cited by 6
- Ideal.ker_quotient_liftproof · cited by 4
- Ideal.ideal_prod_eqproof · cited by 4
- Algebra.Generators.map_ofComp_kerproof · cited by 3
- Ideal.map_eq_submodule_mapproof · cited by 3
- Algebra.Generators.Cotangent.surjective_map_ofCompproof · cited by 3
- Ideal.map_sInfproof · cited by 3
- PowerSeries.exist_eq_span_eq_ncard_of_X_notMemproof · cited by 2
- Ideal.map_fst_prodproof · cited by 1
- image_comap_zeroLocus_eq_zeroLocus_comapproof · cited by 1
- DividedPowers.Quotient.OfSurjective.dividedPowers_uniqueproof · cited by 1