Exercise 15, § 2

Let q be a primary ideal in a ring A and p its radical. Show that in the polynomial ring $B = A[X]$ the ideal $Bq$ is primary and has radical the ideal $Bp$. (Let $f(X) = \sum_j a_j X^j$ and $g(X) = \sum_j b_j X^j$ be two non-constant polynomials such that $fg \in Bq$ and $f \notin Bp$. Let $a_1$ be the coefficient of smallest index not belonging to p; let a be the ideal of A generated by $a_1, a_2, \ldots, a_{m-1}$; there exists an integer k such that $a^k \subset q$. Let $q_i$ be the transporter of $a^{k-i}$ in q for $i \leq k$. Show by induction on i that $g \in Bq_i$, arguing by reductio ad absurdum.)