# Answer on Abstract Algebra Question for Tsit Lam

Question #17264

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

Expert's answer

Say

and similarly

(

So, element a is quasi-regular.

*a*^{n}^{+1}= 0. Then*a ◦*(*−a − a*^{2}*−· · ·−a*)=^{n}*−a*^{2}*−· · ·−a*+^{n}*a*(*a*+*a*^{2}+*· · ·*+*a*) = 0^{n}*,*and similarly

(

*−a − a*^{2}*−· ··−a*)^{n}*◦ a*= 0*.*So, element a is quasi-regular.

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