Reducedness of a formal power series ring
= Reducedness of a formal power series ring
{title2=$R\text{ reduced}\iff R[[X]]\text{ reduced}$}
A <formal power series ring> is a <reduced ring> if and only if its coefficient ring is reduced. A nonzero constant nilpotent proves one direction. In the other direction, a nonzero series with first nonzero coefficient $a_r$ has first coefficient $a_r^m$ in its $m$th power; this is nonzero over a <reduced ring>.