Exercise 24, § 2

In a Laskerian ring A (Exercise 23), let $p_1, p_2, p_3$ be three distinct prime ideals such that $p_1 \subset p_2 \subset p_3$; let $x_2 \in p_2 - p_1, x_3 \in p_3 - p_2$. Replacing if need be $p_2$ and $p_3$ by prime ideals contained in $p_2, p_3$ respectively and containing respectively $x_2$ and $x_3$, we may suppose that $p_2$ is minimal in $\operatorname{Ass}_f(A/(p_1 + Ax_1))$ and $p_3$ is minimal in $\operatorname{Ass}_f(A/p_2 + Ax_3))$. Consider the ideal $a = p_1 + x_2 p_3$.

(a) Show that $x_2 \notin a$ and $x_3^k \notin a$ for every integer $k > 0$; deduce that a is not primary in A. Show that $p_2$ is a minimal element of $\operatorname{Ass}_f(A/a)$.

(b) Let $a = \bigcap_{i=1}^n q_i$ be a reduced primary decomposition of a in A and let $p_i'$ be the radical of $q_i$; suppose that $p_3 \notin p_i'$ for $1 \leq i \leq s, p_3 \subset p_i'$ for $s+1 \leq i \leq n$; show that of necessity $s < n$ and $x_2 \in q_i$ for $1 \leq i \leq s$. Show that there exists an index $i \geq s+1$ such that $p_3 = p_i'$. (Argue by reductio ad absurdum: in the opposite case, there would exist $y \notin p_3$ such that $y \in q_i$ for all $i \geq s+1$; then $x_2 y \in a$ and prove that this relation contradicts $y \notin p_3$.)

Conclude that $p_3$ is a non-minimal element of $\mathrm{Ass}_f(A/a)$; we denote by $q'_3$ the primary component of $a$ in $A$ which corresponds to it in the above decomposition.

(c) Let $b = p_2 \cap q'_3$; on the other hand let $m > 0$ be such that $x_3^m \in q'_3$ and let $c = b + Ax_3^{2m}$. Show that the saturation of $c$ with respect to $p_3$ is a $p_3$-primary ideal $q''_3$ (show that every element of $p_3$ is a power in $q''_3$ by using the fact that the saturation of $p_2 + Ax_3$ with respect to $p_3$ is primary). Prove on the other hand that $x_3^m \notin q''_3$ and that $b = p_2 \cap q''_3$; then there are two distinct reduced primary decompositions of the ideal $b$.