= Principal ideal property of quotients of a Dedekind domain
{title2=$I\ne0\implies R/I\text{ is a principal ideal ring}$}
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 $P^i/P^n$, $0\leq i\leq n$. 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 $r\geq1$. 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>.
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