Ring theory MOC

Dedekind domain

A Dedekind domain 𝑅 is an integral domain such that #m/def/ring

1. 𝑅 is Noetherian;
2. 𝑅 is integrally closed;
3. 𝑅 has Krull dimension 1, i.e. every nontrivial prime ideal is maximal, and there exists a nontrivial prime ideal.

Results

• A Dedekind domain admits UFI, moreover it is a UFD iff it is a PID.
• Fractional ideals of a Dedekind domain form an abelian group
• Ring of integers of a number field form a Dedekind domain and lattice.
• Ideals of a Dedekind domain need at most two generators
• A Dedekind domain with finitely many prime ideals is a UFD
• A Dedekind domain is a UFD iff its ideal class group is trivial

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