Exercise 1, § 2

(a) Show that, if $a$ is an ideal $\neq 0$ in a Dedekind domain $A$, the ring $A/a$ is a principal ideal ring (Algebra, Chapter VII, § 1, Exercise 5) (note that $A/a$ is semi-local and argue as in Proposition 1 of no. 2). Deduce again that every fractional ideal of $A$ is generated by at most two elements (cf. Exercise 11 (a)).

(b) In the polynomial ring $A = k[X, Y]$ over a field $k$, let $m$ be the ideal $AX + AY$. Show that for every integer $n$ the minimum number of generators of the ideal $m^n$ is $n + 1$.