Documentation

Mathlib.RingTheory.Nilpotent.Lemmas

Nilpotent elements #

This file contains results about nilpotent elements that involve ring theory.

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theorem RingHom.ker_isRadical_iff_reduced_of_surjective {R : Type u_1} {S : Type u_3} {F : Type u_4} [CommSemiring R] [CommRing S] [FunLike F R S] [RingHomClass F R S] {f : F} (hf : Function.Surjective ⇑f) :
Ideal.IsRadical (RingHom.ker f) ↔ IsReduced S
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theorem isRadical_iff_span_singleton {R : Type u_1} {y : R} [CommSemiring R] :
IsRadical y ↔ Ideal.IsRadical (Ideal.span {y})
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def nilradical (R : Type u_3) [CommSemiring R] :
Ideal R

The nilradical of a commutative semiring is the ideal of nilpotent elements.

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Instances For
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    theorem mem_nilradical {R : Type u_1} [CommSemiring R] {x : R} :
    x ∈ nilradical R ↔ IsNilpotent x
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    theorem nilradical_eq_sInf (R : Type u_3) [CommSemiring R] :
    nilradical R = sInf {J : Ideal R | Ideal.IsPrime J}
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    theorem nilpotent_iff_mem_prime {R : Type u_1} [CommSemiring R] {x : R} :
    IsNilpotent x ↔ ∀ (J : Ideal R), Ideal.IsPrime J → x ∈ J
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    theorem nilradical_le_prime {R : Type u_1} [CommSemiring R] (J : Ideal R) [H : Ideal.IsPrime J] :
    nilradical R ≤ J
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    @[simp]
    theorem nilradical_eq_zero (R : Type u_3) [CommSemiring R] [IsReduced R] :
    nilradical R = 0
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    @[simp]
    theorem LinearMap.isNilpotent_mulLeft_iff (R : Type u_1) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) :
    IsNilpotent (LinearMap.mulLeft R a) ↔ IsNilpotent a
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    @[simp]
    theorem LinearMap.isNilpotent_mulRight_iff (R : Type u_1) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) :
    IsNilpotent (LinearMap.mulRight R a) ↔ IsNilpotent a
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    @[simp]
    theorem LinearMap.isNilpotent_toMatrix_iff {R : Type u_1} [CommSemiring R] {ι : Type u_3} {M : Type u_4} [Fintype ι] [DecidableEq ι] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (f : M →ₗ[R] M) :
    IsNilpotent ((LinearMap.toMatrix b b) f) ↔ IsNilpotent f
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    theorem Module.End.IsNilpotent.mapQ {R : Type u_1} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {f : Module.End R M} {p : Submodule R M} (hp : p ≤ Submodule.comap f p) (hnp : IsNilpotent f) :
    IsNilpotent (Submodule.mapQ p p f hp)