Dedekind domain

A Dedekind domain is a commutative integral domain satisfying the following properties:

• is a noetherian ring.
• Every prime ideal of is a maximal ideal.
• is integrally closed in its field of fractions.

Dedekind domains are very important in abstract algebra and number theory. For example, the ring of integers of any number field is a Dedekind domain.

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.