11. Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains . 12. An immediate consequence of the definition is that every principal ideal domain ( PID ) is a Dedekind domain . 13. In fact, this is the definition of a Dedekind domain used in Bourbaki's " Commutative algebra ". 14. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective. 15. In fact a Dedekind domain is a unique factorization domain ( UFD ) if and only if it is a PID. 16. All Dedekind domains of characteristic 0 and all local Noetherian rings of dimension at most 1 are J-2 rings. 17. While every number ring is a Dedekind domain , their union, the ring of algebraic integers, is a Pr�fer domain. 18. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. 19. Applying this theorem when " R " is itself a PID gives us a way of building Dedekind domains out of PIDs. 20. There is a version of unique prime factorization for the ideals of a Dedekind domain ( a type of ring important in number theory ).
