Say $X \to Y$ is a surjective map of algebraic varieties, and $Z \subset Y$ is nonreduced. Then is the preimage $Z \times_Y X$ also nonreduced?
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In Allen's notation, take: $R = k[t]$, $X = \operatorname{Spec} R$ $S = k[x,y]/(y^2  x^2(x1) )$, $Y= \operatorname{Spec} S$. with the map defined by: $y = t(t^2+1)$ $x=(t^2+1)$ $I=(x)$, $Z = \operatorname {Spec} S/I$. $S/I$ contains a nilpotent, $y$ so $Z$ is nonreduced. $X \to Y$ is a surjective map of varieties. $X \times_Y Z = \operatorname {Spec} R/RI = \operatorname{Spec} k[t]/(t^2+1)$, which is reduced. 


This is local, of course. So $R \leftarrow S : \phi$ is an injection of domains, and $I \leq S$ is a nonradical ideal; is $R \phi(I)$ a nonradical ideal of $R$? Say $s$ descends to a nonzero nilpotent of $S/I$. Then $\phi(s)$ will likewise be nilpotent in $R / R\phi(I)$, and the injectivity says it will be nonzero. 

