Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Algebra / 3 / c / Solution

A coset is a nilpotent element of the quotient exactly when some positive power is zero: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. ThereforeIn standard terms, an ideal is radical exactly when its quotient is a reduced ring.