Answer to Question #17264 in Abstract Algebra for Tsit Lam

Question

Show that any nilpotent element is quasi-regular in every ring.

Answer 1

Say aⁿ⁺¹= 0. Then
a ◦ (−a − a² −· · ·−aⁿ)= −a² −· · ·−aⁿ + a(a + a²+ · · · + aⁿ) = 0,
and similarly
(−a − a² −· ··−aⁿ) ◦ a = 0.
So, element a is quasi-regular.