Nilpotent ideal with semisimple quotient radical criterion
= Nilpotent ideal with semisimple quotient radical criterion
{title2=$I^N=0,\ J(A/I)=0\Rightarrow I=J(A)$}
If $I$ is a <nilpotent ideal> and $J(A/I)=0$, then $I=J(A)$. Every $1-ax$, $x\in I$, has a finite geometric-series inverse, giving $I\subseteq J(A)$. The radical's image lies in $J(A/I)=0$, giving the reverse inclusion. A <semisimple algebra> as quotient supplies the required zero radical.