Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 1 / 9E / i / Solution

If $a^m=b^n=0$, then every term in $(a+b{)}^{m+n-1}$ vanishes; also $(ra{)}^m=0$. Thus the nilpotents form an ideal $N$. The quotient $R/N$ has no nonzero nilpotents.