Theorems · Theorem · ring theory
exists_isIdempotentElem_mul_eq_zero_of_ker_isNilpotent_aux
∀ {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst_1 : Ring S] (f : R →+* S),
(∀ x ∈ RingHom.ker f, IsNilpotent x) →
∀ e₁ ∈ f.range,
IsIdempotentElem e₁ →
∀ (e₂ : R), IsIdempotentElem e₂ → e₁ * f e₂ = 0 → ∃ e', IsIdempotentElem e' ∧ f e' = e₁ ∧ e' * e₂ = 0- Defined in
- Mathlib.RingTheory.Idempotents
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites34
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
- RingHomstatement and proof · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Idealstatement · cited by 4,748
- Nontrivialproof · cited by 2,416
- MulZeroClass.mul_zeroproof · cited by 2,091
- mul_assocproof · cited by 1,667
- map_mulproof · cited by 1,137
- MulOppositeproof · cited by 1,135
- pow_zeroproof · cited by 1,094
- sub_selfproof · cited by 996
- sub_zeroproof · cited by 938
Cited by1
Results whose statement or proof uses this declaration.
- exists_isIdempotentElem_mul_eq_zero_of_ker_isNilpotentproof · cited by 2