| 1. | It is also universally catenary as it is a Dedekind domain.
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| 2. | Some authors add the requirement that a Dedekind domain not be a field.
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| 3. | In Dedekind domains, the situation is much simpler.
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| 4. | These are the prototypical examples of Dedekind domains.
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| 5. | This notion can be used to study the various characterizations of a Dedekind domain.
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| 6. | Now let M be a finitely generated module over an arbitrary Dedekind domain R.
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| 7. | For a Dedekind domain this is of course the same as the ideal class group.
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| 8. | Over an arbitrary Dedekind domain one has
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| 9. | In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain.
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| 10. | On the other hand, the ring of integers in a number field is always a Dedekind domain.
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