case intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
n : ℕ
hn : Ideal.radical ⊥ ^ n ≤ ⊥
⊢ IsNilpotent (nilradical R)
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
⊢ IsNilpotent (nilradical R)
obtain ⟨n, hn⟩ := Ideal.exists_radical_pow_le_of_fg (⊥ : Ideal R) (IsNoetherian.noetherian _)
no goals
case intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
n : ℕ
hn : Ideal.radical ⊥ ^ n ≤ ⊥
⊢ IsNilpotent (nilradical R)
exact ⟨n, eq_bot_iff.mpr hn⟩