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