Exercise 3, § 6

Let $K$ be a field, $B$ the polynomial ring $K[X_1, X_2]$ and $p$ the ideal of $B$ generated by the polynomial $X_1^3 - X_2^2$.

(a) Show that $p$ is prime and therefore $A = B/p$ is an integral domain. Let $a$ and $b$ denote the canonical images of $X_1$ and $X_2$ in $A$; $a$ and $b$ are not invertible in $A$ and $a^3 = b^2$.

(b) Let $m$ be the (maximal) ideal of $A$ generated by $a$ and $b$. Show that the symmetric algebra $S(m)$ is not an integral domain. (Observe that $m$ is identified with the quotient module $A^2/N$, where $N$ is the submodule of $A^2$ generated by $(b, -a)$; conclude that $S(m)$ is isomorphic to the quotient ring $A[U, V]/(bU - aV)$ and show that in the polynomial ring $A[U, V]$ the principal ideal $(bV - aU)$ is not prime, by considering the product $b^2(bV^3 - U^3)$.)