Exercise 13, § 4

Let $A$ be a noetherian integral ring, $K$ its field of fractions, $\overline{A}$ its integral closure, $f$ a nonzero element of A, $\mathfrak{P}$ a prime ideal of height 1 of $\overline{A}$ containing $f$, $p$ the prime ideal

$^1$ For every ring B, we denote by $B_{\text{red}}$ the quotient of B by the ideal of nilpotent elements of B.

$A \cap \mathfrak{P}$ of A. Prove that one has $p \in \operatorname{Ass}_A(A/fA)$ by reducing to the case where A is local, with maximal ideal $p$, and by establishing successively under this assumption the following assertions: a) $\mathfrak{P}$ is divisoriel (use Exer. 10, c)). b) $p$ is divisoriel (use Exer. 12, f)). c) If $a_1, ..., a_r$ are nonzero elements of K such that

and there exists an index $i$ such that $p = A \cap a_i^{-1}A$. d) There exists a nonzero element $g$ of A such that $p \in \operatorname{Ass}_A(A/gA)$. e) $p \in \operatorname{Ass}_A(A/fA)$.