Dedekind domain

Remarks

Example: $\Z$ is a Dedekind domain. It is a principal ideal domain, hence Noetherian. By the rational root test, a rational root of a monic integer polynomial is an integer. Every nonzero prime ideal $(p)$ is maximal, since $\Z/(p)$ is a field. More generally, the ring of integers of every number field is a Dedekind domain, for example $\Z[\sqrt{-5}]$, which is not a unique factorisation domain. Non-example: $\Z[\sqrt{-3}]$ fails clause 2, since $\omega = (1 + \sqrt{-3})/2$ lies in its field of fractions, is a root of $x^2 - x + 1$, and is not in $\Z[\sqrt{-3}]$. Another non-example: $k[x, y]$ for a field $k$ fails clause 3, since $(x)$ is a nonzero prime ideal contained in the larger ideal $(x, y)$. For a Noetherian integral domain $R$ that is not a field, each of these conditions is equivalent to being Dedekind. First: every nonzero proper ideal is a product of prime ideals, in a unique way up to order. Second: every nonzero fractional ideal of $R$ is invertible. Some authors also count fields as Dedekind domains.