- is a noetherian ring.
- Every prime ideal of is a maximal ideal.
- is integrally closed in its field of fractions.
There are several very nice properties of Dedekind domains:
- Dedekind domains have unique prime factorizations of ideals (but not necessarily of elements).
- Ideals are invertible if we extend to fractional ideals. Let be a Dedekind domain with field of fractions , and let be any nonzero ideal of . Then set . We call an ideal invertible if . (Note that this is always a subset, but it is not always equal unless we are in a Dedekind domain.) In fact, the converse is true as well: if all nonzero ideals are invertible, then is a Dedekind domain. This is sometimes used as a definition.
There are also various properties of homological importance that Dedekind domains satisfy.