Dedekind domain

Symbol-free definition
An integral domain is termed a Dedekind domain if it satisfies the following equivalent conditions:


 * It is a Noetherian normal domain of Krull dimension 1
 * Every nonzero ideal is invertible in the field of fractions and can be expressed uniquely as a product of prime ideals

Conjunction with other properties
Any unique factorization domain which is also a Dedekind domain, is also a principal ideal domain.

Module theory
Any finitely generated module $$M$$ over a Dedekind domain $$R$$ can be expressed as a direct sum as follows:

$$M \cong R/I_1 \oplus R/I_2 \oplus \ldots \oplus R/I_n$$

where $$I_1 \subset I_2 \subset \ldots \subset I_n$$ is an ascending chain of ideals, which could reach $$R$$.