Let $A$ be a commutative ring with unit. Show that $A$ is reduced iff for every prime ideal $\mathfrak{p}\subseteq A$, $A_{\mathfrak{p}}$ is reduced.
This corresponds more or less to exercise 5, chapter 3 of Atiyah-Macdonald. It is useful to remember that $\text{Nil}(A_{\mathfrak{p}})=(\text{Nil}(A))_{\mathfrak{p}}$ (*).
($\Rightarrow$) Obvious by (*).
($\Leftarrow$) By the fact that "being $0$" is a local property which is satisfied by $\text{Nil}(A)$, again because of (*).
I wonder if the argument for the $\Leftarrow$ is sufficient.