Dedekind domain (reviewed)

A Dedekind domain $D$ is a integral domain which is not a field such that

1. $D$ is Noetherian;
2. every non-zero prime ideal is maximal;
3. $D$ is integrally closed.

The ring of integers of a number field is always a Dedekind domain, as is every discrete valuation ring.

In a Dedekind domain, every non-zero ideal $I$ can be written as a product of non-zero prime ideals, $$I=P_1P_2\cdots P_k,$$ and the product is unique up to the order of the factors. Repeated factors are often grouped, so we write $I=Q_1^{e_1}\cdots Q_g^{e_g}$ where the $Q_i$ are non-zero prime ideals of $D$.

In addition, every fractional ideal $I$ is invertible in the sense that there exists a fractional ideal $J$ such that $IJ=D$.