Exercise 25, § 2

Let $A$ be a Laskerian ring. Let $a$ be an ideal of $A$ and $a = \bigcap_{1 \leq i \leq n} q_i$ a reduced primary decomposition of $a$ in $A$; let $p_i$ denote the radical of $q_i$; suppose that the minimal elements of $\mathrm{Ass}_f(A/a)$ are the $p_i$ such that $1 \leq i \leq s$ and that $s < n$. We write $b = \bigcap_{1 \leq i \leq s} q_i, c = \bigcap_{s+1 \leq i \leq n} q_i$.

(a) Show that there exists $x \in c$ such that $x \notin p_i$ for $1 \leq i \leq s$ and that $a = b \cap (a + Ax)$.

(b) Suppose for example that $p_{s+1}$ is minimal in $\mathrm{Ass}_f(A/c)$. Show that $xp_{s+1} + a \neq Ax + a$ and $a = b \cap (xp_{s+1} + a)$; moreover, $p_{s+1}$ is minimal in $\mathrm{Ass}_f(A/b)$, where $b = xp_{s+1} + a$. Show that the saturation of $b$ with respect to $p_{s+1}$ is different from $q''_1$. (Note that this saturation cannot contain $x$.)