Dedekind domain

A Dedekind domain is a Noetherian integrally closed domain of Krull dimension one. Equivalently, every localization at a nonzero prime ideal is a discrete valuation ring.

For a Dedekind domain $R$ and a nonzero proper ideal $I$, the quotient ring $R/I$ is a principal ideal ring. First, the correspondence theorem for ideals and unique factorization of ideals in a Dedekind domain show that the ideals of $R/P^n$ are exactly $P^i/P^n$, $0\le i\le n$. Consecutive powers are distinct by cancellation of invertible fractional ideals. Choose $\pi \in P^i\setminus P^{i+1}$. There is no intermediate ideal between these consecutive powers, so $P^i=(\pi )+P^{i+1}$. Multiplication by $P^r$ and substitution show inductively that $P^i=(\pi )+P^{i+r}$ for every $r\ge 1$. Taking $r=n-i$ gives a single generator modulo $P^n$. Now factor $I$ into pairwise comaximal ideals $P_j^{n_j}$. The Chinese remainder theorem writes $R/I$ as a finite direct product of rings of the preceding type, which is a principal ideal ring. It need not be an integral domain, as $\mathbb{Z}/4\mathbb{Z}$ demonstrates. Applying the result to $(a)$ for any $0\ne a\in I$ makes $I/(a)$ cyclic, yielding two generators for an ideal in a Dedekind domain.

Every ideal of a Dedekind domain is generated by at most two elements. Given a nonzero ideal $I$ and any prescribed nonzero $a\in I$, the Chinese remainder theorem supplies $b\in I$ with $v_P((b))=v_P(I)$ at every prime ideal dividing $(a)$. The prime-ideal valuation in a Dedekind domain then gives $I=(a,b)$.

Every nonzero proper ideal of a Dedekind domain has a unique factorization, up to reordering, as a finite product of positive powers of distinct nonzero prime ideals. The unit ideal is the empty product. This is distinct from unique factorization of elements.

For a nonzero ideal $I$ in a Dedekind domain, $v_P(I)$ is the exponent of the nonzero prime ideal $P$ in the unique factorization of ideals in a Dedekind domain. Then $I\subseteq J$ if and only if $v_P(I)\ge v_P(J)$ for every $P$, and $v_P(I+J)=\min (v_P(I),v_P(J))$.

The maximal ideal of an integrally closed Noetherian local domain of Krull dimension one is principal. Hence every such local ring is a discrete valuation ring.