Theorems · Theorem · commutative algebra
IsNilpotent.map_iff
∀ {R : Type u_1} {S : Type u_2} [inst : MonoidWithZero R] [inst_1 : MonoidWithZero S] {r : R} {F : Type u_3}
[inst_2 : FunLike F R S] [MonoidWithZeroHomClass F R S] {f : F},
Function.Injective ⇑f → (IsNilpotent (f r) ↔ IsNilpotent r)- Defined in
- Mathlib.RingTheory.Nilpotent.Defs
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses Classical.choice
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 and proof · cited by 62,936
- FunLikestatement and proof · cited by 2,560
- map_powproof · cited by 503
- MonoidWithZerostatement and proof · cited by 456
- IsNilpotentstatement and proof · cited by 248
- map_eq_zero_iffproof · cited by 62
- MonoidWithZeroHomClassstatement and proof · cited by 37
- IsNilpotent.mapproof · cited by 12
Cited by9
Results whose statement or proof uses this declaration.
- Module.End.disjoint_genEigenspaceproof · cited by 7
- PrimeSpectrum.mem_image_comap_basicOpenproof · cited by 3
- LieModule.isNilpotent_derivedSeries_of_traceForm_eq_zeroproof · cited by 1
- isNilpotent_tensor_residueField_iffproof · cited by 1
- Polynomial.mem_image_comap_C_basicOpenproof · cited by 1
- IsReduced.tensorProduct_of_flat_of_forall_fgproof · cited by 1
- MvPolynomial.mem_image_comap_C_basicOpenproof · cited by 1
- PrimeSpectrum.isOpenMap_comap_algebraMap_tensorProduct_of_fieldproof · cited by 0
- MvPolynomial.isNilpotent_iffproof · cited by 0