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when is u+v nilpotent for nilpotent u,v

Source: VJIMC 2006 2.1

June 28, 2021

Ring Theoryabstract algebra

Problem Statement

(a) Let

u

and

v

be two nilpotent elements in a commutative ring (with or without unity). Prove that

u+v

is also nilpotent. (b) Show an example of a (non-commutative) ring

R

and nilpotent elements

u,v\in R

such that

u+v

is not nilpotent.