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Rings, Modules, Fields

9 Dedekind domains

Often we don’t have unique factorization of elements in our ring. For example, the classic example is \(\ZZ [\sqrt {-5}]\), where \((1+\sqrt {-5})*(1-\sqrt {5}) = 2*3\), but in this case, and many others, we have the next nicest thing: unique factorization of ideals.
  • Definition 9.1. A Dedekind domain is a domain with factorization of nonzero ideals in prime ideals.

  • Lemma 9.2. \(\pp |a\) iff \(\pp |a\) in the monoid of ideals under multiplication

  • Lemma 9.3. Factorization in a Dedekind domain is unique.

  • Proof.