Exercise 12, § 2

Let $A$ be an integral domain and $K$ its field of fractions. Show that the following properties are equivalent: (a) For every prime ideal $p$ of $A$, the local ring $A_p$ is a valuation ring. (b) For every maximal ideal $m$ of $A$, the local ring $A_m$ is a valuation ring.

(y) Every finitely generated ideal $\neq 0$ in $\mathbf{A}$ is invertible (and therefore the finitely generated fractional ideals $\neq 0$ form a group).

(6) Every torsion-free finitely generated $\mathbf{A}$-module is projective.

(E) For all $x \in K$, the fractional ideal $\mathbf{A} + \mathbf{A}x$ is invertible.

($\zeta$) For all $x \neq 0$ in $K$, there exists $y \in K$ such that $y \equiv 0 \pmod{1}$, $y \equiv 0 \pmod{x}$ and $y \equiv 1 \pmod{1-x}$.

($\eta$) If $(a_i)$ is a finite family of ideals of $A$, for the system of congruences $x \equiv c_i \pmod{a_i}$ to have a solution it is necessary and sufficient that $c_i \equiv c_j \pmod{(a_i + a_j)}$ for every pair of indices $i, j$ ("Chinese remainder theorem").

($\theta$) If $a, b, c$ are three ideals of $\mathbf{A}$, then $a \cap (b + c) = a \cap b + a \cap c$.

(t) If $a, b, c$ are three ideals of $\mathbf{A}$, then $a + (b \cap c) = (a + b) \cap (a + c)$.

(x) $\mathbf{A}$ is integrally closed and every finitely generated ideal of $\mathbf{A}$ is divisorial.

($\lambda$) $\mathbf{A}$ is integrally closed and for $z \neq 0$ in $K$ there exist $x, y$ in $\mathbf{A}$ such that $z = x + yz^2$.

(p) $\mathbf{A}$ is integrally closed and, if $a, b, c$ are three finitely generated fractional ideals $\neq 0$, the relation $ab = ac$ implies $b = c$.

(v) For every finitely generated $\mathbf{A}$-module $M$, the torsion submodule of $M$ is a direct factor of $M$.

($\sigma$) Every ring $B$ such that $\mathbf{A} \subset B \subset K$ is an integrally closed domain.

Then $\mathbf{A}$ is called a Prüferian domain (or a Prüfer domain).

(To prove the equivalence of ($a$), ($\beta$), (y) and ($\delta$), use Chapter II, § 5, no. 2, Theorem 1 and no. 6, Theorem 4. To prove that ($E$) implies (y), use the identity $(a + b)(b + c)(c + a) = (a + b + c)(ab + bc + ca)$ between three fractional ideals in an integral domain. To prove that ($\zeta$) implies ($\eta$), argue by induction on the number of $a_i$. The equivalence of (q), (l) and (t) has been shown in Algebra, Chapter VI, § 1, Exercise 25. Prove that (A) implies ($E$) by noting that ($\lambda$) implies the relation

using Exercise 6 (b) of § 1 and noting that

To prove that (p) implies (A) noting that always

Prove that (x) implies ($\lambda$) noting that every fractional principal ideal $\mathbf{A}$ which contained $\mathbf{A}$ and $\mathbf{A}z^2$ also contains $\mathbf{A}z$. To prove that (v) implies (y), consider, for a finitely generated integral ideal $a \neq 0$ of $\mathbf{A}$, a non-zero element $c \in a(A : a)$ and apply Exercise 32 of § 1 to the ideal $b = ca$. To prove that ($s$) implies ($\beta$), reduce it to the case where $\mathbf{A}$ is a local ring, consider an element $z \in K - \mathbf{A}$ and show that $z^{-1} \in \mathbf{A}$, noting that the hypothesis implies that $z \in \mathbf{A}[z^2]$.