nilpotencyClass — Mathlib

English

If u is a unit and r commutes with u, then IsNilpotent((u)·r) ↔ IsNilpotent r; equivalently, multiplication by a unit preserves nilpotency.

Русский

Если u — единица и r коммутирует с u, то IsNilpotent(u·r) ⇔ IsNilpotent r; то есть умножение на единицу сохраняет нилпотентность.

LaTeX

$$$ [MonoidWithZero\\ R] \\{r\\ u : R\\} (hu : \\IsUnit u) (h_comm : Commute r u) : IsNilpotent (u * r) \\iff IsNilpotent r $$$

Lean4

/-- If `x` is nilpotent, the nilpotency class is the smallest natural number `k` such that
`x ^ k = 0`. If `x` is not nilpotent, the nilpotency class takes the junk value `0`. -/
noncomputable def nilpotencyClass : ℕ :=
  sInf {k | x ^ k = 0}

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