Exercise 6, § 1

Let $A$ be a commutative ring and $B$ a commutative $A$-algebra; suppose that the minimal prime ideals of B are finite in number; let $p_i$ ($1 \leq i \leq n$) be these ideals (note that this hypothesis holds if B is Noetherian). For an element $x \in B$ to be integral over A, it is necessary and sufficient that each of its canonical images $x_i$ in $B/p_i$ be integral over A. (If this condition holds, show first that $x$ is integral over the subalgebra of B generated by the nilradical $\mathfrak{N} = \bigcap_i p_i$ of B and use the fact that the elements of $\mathfrak{N}$ are nilpotent.)