Exercise 1, § 1

Let $K$ be a field and $A$ the subring of the ring of formal power series $K[[T]]$ consisting of the series in which the coefficient of $T$ is 0; $A$ is a Noetherian local integral domain which is not integrally closed.

(a) Show that every principal ideal of $A$, which is non-zero and distinct from $A$, contains a unique element of the form $T^n + \lambda T^{n+1}$ where $\lambda \in K$, $n \geq 2$; moreover, this ideal contains all the powers $T^m$ where $m \geq n + 2$. Deduce that the non-principal ideals of $A$ are the ideals $AT'' + AT''+1$ where $n \geq 2$.

(b) Show that every fractional ideal of $A$ is divisorial (prove that every non-principal ideal is the intersection of two principal ideals). Describe the structure of the monoid $D(A)$. The only prime ideals of $A$ are the maximal ideal $m = AT^2 + AT^3$ and $(0)$.