Exercise 3, § 1

Let $A$ be an integral domain in which every nonempty set of ideals has a maximal element and in which every maximal ideal is principal ; show that $A$ is a principal ideal domain. (Notice that if $a \neq A$ is an ideal of $A$ then there exists a maximal ideal $(p) \supset a$ such that $\frac{1}{p} a$, which is an ideal of $A$, strictly contains $a$.)

b) Let $K$ be a field, let $K_1 = K((X))$ be the field of formal power series in one indeterminate $X$ over $K$ (IV, p. 38), and let $B = K_1[[Y]]$ be the ring of formal power series in $Y$ with coefficients in $K_1$. Let $A$ be the subring of $B$ consisting of those power series $\sum_{n=0}^\infty a_n(X) Y^n$ such that the power series $a_0(X)$ involves only powers of $X$ with exponent $\geq 0$. Show that the principal ideal $(X)$ is the unique maximal ideal of $A$, but the ideal of $A$ generated by the elements $Y X^{-n}$ ($n \geq 0$ an arbitrary integer) is not principal