Suppose we have a ring homomorphism $\phi:A\rightarrow B$ and an ideal $J\subseteq B$.
I just spent way too much time on an exercise in commutative algebra, because the element-free definition of the radical of $J$ as $$\sqrt{J} = \bigcap \limits_{q\in \operatorname{Spec}B, J\subseteq q} q$$ suggested to me that there might be a discrepancy between $$\phi^{-1}(\sqrt{J}) = \bigcap \limits_{q \in \operatorname{Spec} B, J \subseteq q} \phi^{-1}(q)$$ and $$\sqrt{\phi^{-1}(J)} = \bigcap \limits_{p\in \operatorname{Spec} A, \phi^{-1}(J)\subseteq p} p$$ (the latter intersection appears to be smaller in total)
But using the element-wise definition of a radical in terms of powers one can deduce that both sets always coincide! So I am wondering, whether I should have noticed it somehow while working with the element-free definition…