Equivalent local characterizations of Dedekind domains

If $R$ is Dedekind, then [L1] gives that each localisation at a nonzero prime is a DVR. This proves $(1)\Rightarrow (2)$.

Assume (2). Then [L2] makes $R$ integrally closed, and [L3] gives $\dim R=1$ together with maximality of the nonzero primes. Because $R$ was assumed Noetherian and nonfield, [F1] now says that $R$ is Dedekind. This proves $(2)\Rightarrow (1)$.

Still assuming (2), let $I$ be a nonzero proper ideal and let $\mathfrak{m}$ be a maximal ideal. If $I⊈\mathfrak{m}$, then $I_{\mathfrak{m}}=R_{\mathfrak{m}}$ because some element of $I$ becomes a unit. If $I\subseteq \mathfrak{m}$, then $\mathfrak{m}$ is nonzero and [L4] applies in the DVR $R_{\mathfrak{m}}$ from (2), so $I_{\mathfrak{m}}$ is principal. Therefore (3) holds.

Assume (3), and let $\mathfrak{m}$ be a maximal ideal. By [L6] and [L7], the ring $R_{\mathfrak{m}}$ is a Noetherian local domain. Let $J\subset R_{\mathfrak{m}}$ be a nonzero proper ideal, and let $I:=\{r\in R:r\mathrm{/}1\in J\}$. Then $J=I{R}_{\mathfrak{m}}$, and $I$ is a nonzero proper ideal contained in $\mathfrak{m}$. Applying (3) at the maximal ideal $\mathfrak{m}$ shows that $J=I_{\mathfrak{m}}$ is principal. Hence every nonzero proper ideal of the local domain $R_{\mathfrak{m}}$ is principal, and because $R$ is not a field its maximal ideal $\mathfrak{m}{R}_{\mathfrak{m}}$ is nonzero. Therefore [L5] makes $R_{\mathfrak{m}}$ a DVR. Now let $\mathfrak{p}$ be a nonzero prime ideal and choose a maximal ideal $\mathfrak{m}\supseteq \mathfrak{p}$. In the DVR $R_{\mathfrak{m}}$, the extended ideal $\mathfrak{p}{R}_{\mathfrak{m}}$ is a nonzero prime ideal, so it equals the maximal ideal $\mathfrak{m}{R}_{\mathfrak{m}}$. Contracting to $R$ gives $\mathfrak{p}=\mathfrak{m}$, so every nonzero prime is maximal and therefore $R_{\mathfrak{p}}$ is a DVR by the first part applied with $\mathfrak{m}=\mathfrak{p}$. This proves $(3)\Rightarrow (2)$.

By [L3], any of the equivalent conditions forces every nonzero prime ideal of $R$ to be maximal.