Let $R$ be an $S$-algebra and let $I$ be a primary ideal such that $R/\sqrt{I}$ is flat over $S$ where $\sqrt{I}$ is the radical of $I$. Is $R/I$ flat over $S$? I don't mind assuming $R,S$ Noetherian, and even regular.
(as a reason for why one might be interested in this question, consider an ideal $J \subset B$ and let $I$ be the unique primary ideal in a primary decomposition corresponding to a minimal associated prime of $J$. In this case $I$ endows the corresponding irreducible component $Z=V(I)$ of $X=\mathrm{Spec} B/J$ with canonical scheme structure. The statement above says that if $Z_{red}$ is flat over $Y=\mathrm{Spec} A$, then $Z$ itself is flat over $Y$)