IsNilpotent — Mathlib

English

If there exists n with x^n = 0, then x is nilpotent.

Русский

Если существует n such что x^n = 0, то x нилпотентен.

LaTeX

$$$\\exists n:\\mathbb{N},\\ x^n=0 \\Rightarrow \\IsNilpotent(x)$$$

Lean4

/-- An element is said to be nilpotent if some natural-number-power of it equals zero.

Note that we require only the bare minimum assumptions for the definition to make sense. Even
`MonoidWithZero` is too strong since nilpotency is important in the study of rings that are only
power-associative. -/
def IsNilpotent [Zero R] [Pow R ℕ] (x : R) : Prop :=
  ∃ n : ℕ, x ^ n = 0

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