Exercise 22, § 2

(a) Let $\mathbf{A}$ be a Noetherian domain each of whose prime ideals $\neq 0$ is maximal, so that every ideal $a \neq 0$ may be written uniquely as a product $\prod_i q_i$ of primary ideals relative to the distinct prime ideals $p_i$ containing $a$.

Let $a$ be an invertible ideal in $\mathbf{A}$ and let $b$ be any ideal $\neq 0$ in $\mathbf{A}$; show that there exists an ideal $c$ such that $b + c = \mathbf{A}$ and $ac$ is principal (Observe that, if $(p_i)_{1 \leq i \leq n}$ is a finite family of distinct maximal ideals of $\mathbf{A}$ and we write $r_i = \prod_{j \neq i} p_j$, then $a = \sum_i a r_i$ and $\bigcap_i a r_i = a p_1 p_2 \ldots p_n$; using Chapter II, § 1, no. 2, Proposition 6, deduce the existence of an element $x \in a$ such that $x^{-1}a + b = \mathbf{A}$.) Deduce that $a$ is generated by two elements (cf. Exercise 1).

(b) Let $K$ be a field and $\mathbf{A}$ the subring of the ring of formal power series $K[[T]]$ consisting of the series $a_0 + T^n P(T)$, where $a_0 \in K, P(T) \in K[[T]]$, for a given integer $n$. Show that $\mathbf{A}$ is a Noetherian local integral domain with a single prime ideal $m \neq 0$, but that the smallest cardinal of a system of generators of $m$ is $n$.