= Solution
A coset is a <nilpotent element> of the quotient exactly when some positive power is zero:
$$
x+J\text{ nilpotent}\ \Longleftrightarrow\ (x+J)^n=J\text{ for some }n\geq1
\ \Longleftrightarrow\ x\in\sqrt J.
$$
If $J=\sqrt J$, this forces every nilpotent coset to equal zero. Conversely, if there is no nonzero nilpotent coset, every $x\in\sqrt J$ has $x+J=0$, hence $x\in J$. Since $J\subseteq\sqrt J$ always, equality follows. Therefore
$$
\boxed{J=\sqrt J\quad\Longleftrightarrow\quad R/J\text{ has no nonzero nilpotent elements}.}
$$
In standard terms, an <ideal> is radical exactly when its quotient is a <reduced ring>.
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