\(R/ \nilrad{R}\) has no nonzero nilpotent elements.
Solution
\[ a + \nilrad{R} \text{ nilpotent } &\implies (a+ \nilrad{R})^n \definedas a^n + \nilrad{R}= \nilrad{R} \\ &\implies a^n \in \nilrad{R} \\ &\implies \exists \ell \text{ such that } (a^n)^\ell = 0 \\ &\implies a\in \nilrad{R} . \]