Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 2 / 11E / a / Solution

For nilpotent elements $a,b$ of the commutative ring, choose positive integers $r,s$ with $a^r=b^s=0$. The binomial theorem giveseach term has either $j\ge r$ or $r+s-1-j\ge s$. Also $(-a{)}^r=0$, $0\in N$, and $(ta{)}^r=t^r{a}^r=0$ for every $t\in R$. Thus $N$ is an additive subgroup stable under multiplication by arbitrary ring elements:Commutativity is essential to the simple binomial argument; an arbitrary sum of nilpotent elements in a noncommutative ring need not be nilpotent.