Exercise 10, § 4

Let $A$ be a ring in which the set of principal ideals is totally ordered by inclusion.

(a) Show that $A$ is a local ring, in which the nilradical $\mathfrak{N}$ is a prime ideal. Show that, either $\mathfrak{N}^2 = \mathfrak{N}$, or $\mathfrak{N}$ is nilpotent; the ring $A/\mathfrak{N}$ is a valuation ring.

(b) If $\mathscr{L}$ is the totally ordered set of principal ideals of $A$, show that the set of ideals of $A$ is totally ordered by inclusion and is isomorphic to the completion of the set $\mathscr{L}$ (Set Theory, Chapter III, § 1, Exercise 15).

(c) Suppose that $\mathfrak{N}^2 = 0$; then $\mathfrak{N}$ may be considered as a module over the valuation ring $V = A/\mathfrak{N}$; let $K$ be the field of fractions of $V$ and $\Gamma$ the order group of a valuation $v$ on $K$ corresponding to $V$; show that there exists in $\Gamma$ two major sets $M \subset M'$ such that $\mathfrak{N}$ is a $V$-module isomorphic to $a(M')/a(M)$ (notation of § 3, no. 4).

(d) Conversely, given a valuation ring $V$ with field of fractions $K$ and order group $\Gamma$ and two major sets $M \subset M'$ in $\Gamma$, let $Q = a(M')/a(M)$ and define on the product additive group $A = V \times Q$ a multiplication by setting

show that in $A$ the set of principal ideals is totally ordered by inclusion; the set $\mathfrak{N}$ of ordered pairs $(0, t)$ where $t \in Q$ is the nilradical of $A$, $\mathfrak{N}^2 = 0$ and $A/\mathfrak{N}$ is isomorphic to $V$.

(e) For every prime number $p$, the ring $A = \mathbf{Z}/p^2\mathbf{Z}$ is such that the set of principal ideals of $A$ is totally ordered by inclusion and the nilradical $\mathfrak{N}$ of $A$ is such that $\mathfrak{N}^2 = 0$; but show that the ring $A$ is not isomorphic to the ring constructed (starting with $V = A/\mathfrak{N}$ and $Q = \mathfrak{N}$) by the method in (d) (note that $A$ contains no subring isomorphic to $\mathbf{Z}/p\mathbf{Z}$).

(f) Show that for every prime number $p$, the algebra $A$ of the group $U_p$ (Algebra, Chapter VII, § 2, Exercise 3) with respect to the prime field $\mathbf{F}_p$ is a ring in which the principal ideals form a set which is totally ordered by inclusion and the nilradical $\%$ satisfies $\%^2 = \%$, but $A$ is not isomorphic to the quotient of a valuation ring by an ideal.