Exercise 2, § 3

(a) Let $A$ be a (commutative) local ring whose maximal ideal $m$ is principal and such that $\bigcap_{n \geq 1} m^n = 0$ (cf. Chapter III, § 3, no. 2, Corollary to Proposition 4). Show that the only ideals $\neq 0$ and $\#A$ of $A$ are the powers $m^n$; deduce that $A$ is Noetherian and is either a discrete valuation ring (Chapter VI) or a quasi-principal ring in which 1 is an indecomposable idempotent (Algebra, Chapter VII, § 1, Exercise 6).

(b) Let $A$ be the ring of germs at the point $t = 0$ of real-valued functions defined and continuous in a neighbourhood of 0 and differentiable at the point 0 (General Topology, Chapter I, 3rd ed., § 6, no. 10). Show that $A$ is a local ring whose maximal ideal $m$ is generated by the germ of the function $j : t \mapsto t ; m^n \neq 0$ for all $n$ but $A$ is not a principal ideal domain. If $c$ is the germ of the function $t \mapsto \exp(-1/t^2)$ and $p$ a prime ideal of $A$ not containing $c$, show that the quotient ring $B = A/p$ is a local integral domain which is not a principal ideal domain and in which the maximal ideal is principal.