Exercise 10, § 2

(a) Let K be a commutative field, A the polynomial ring K[X, Y], $p = (X), q = (Y)$ and S the complement of $p \cup q$ in A. Show that the saturation of $p + q$ with respect to S is distinct from the sum of the saturations of $p$ and $q$.

(b) Let K be a commutative field, A the polynomial ring K[X, Y, Z], $p = (X) + (Z), q = (Y) + (Z)$ and S the complement of $p \cup q$ in A. Show that the saturation of $pq$ with respect to S is distinct from the product of the saturations of $p$ and $q$.