Exercise 15, § 1

(a) Let A be an integral domain and $a, b$ two elements $\neq 0$ of A such that $Aa \cap Ab = Aab$. Show that in the polynomial ring $A[X]$ the principal ideal generated by $aX + b$ is prime (prove that every polynomial $f(X) \in A[X]$ such that $f(-b/a) = 0$ is a multiple of $aX + b$, arguing induction on the degree of $f$).

(b) Let A be a Krull domain and $a, b$ two elements of A such that Aa and Aa + Ab are prime and distinct. Show that $A[X]/(aX + b)$ is a Krull domain and that $C(A[X]/(aX + b))$ is isomorphic to $C(A)$. (Note, with the aid of (a), that $A[X]/(aX + b)$ is isomorphic to $A[b/a] = B$; show that Ba is prime in B. Observe that $A[a^{-1}] = B[a^{-1}]$ is a Krull domain and use Exercise 12.)

(c) Let A be an integrally closed Noetherian domain and $a, b$ two elements of the Jacobson radical r of A. Suppose that the ideal Aa + Ab is a non-divisorial prime ideal; show that Aa and Ab are prime ideals. (Prove first that $Aa \cap Ab = Aab$ considering the ideals of $Ass(A/Aa)$. Show secondly that the relations $xy \in Ab$ and $y \notin Aa + Ab$ imply $x \in Ab + Aa^h$ for all $h$, by induction on $h$; deduce that then $x \in Ab$. Finally, deduce that, if $xy \in Ab, x \notin Ab, y \notin Ab$, then necessarily $x \in Ab + Aa^h$ and $y \in Ab + Aa^h$ for all $h$, arguing by induction on h; deduce a contradiction.)