A Dedekind domain is a Noetherian ring that is an integrally closed domain of Krull dimension one. Equivalently, for a Noetherian domain that is not a field, every localization at a prime ideal at a nonzero prime is a discrete valuation ring. The Noetherian and nonfield qualifications belong to this local characterization; a field would satisfy its condition vacuously.
The ring of integers of a number field is a standard example. Every nonzero proper ideal has a unique factorization into nonzero prime ideals, even when element factorization is not unique. Locally, the exponent of each prime is the discrete valuation of that ideal in its corresponding discrete valuation ring. This connects unique factorization of ideals in a Dedekind domain with local arithmetic.
A **Dedekind domain** is a specific type of ring that plays a significant role in number theory, algebraic geometry, and algebraic number theory. A Dedekind domain is defined as an integral domain that satisfies certain properties. Here are the key characteristics of a Dedekind domain: 1. **Noetherian**: The ring is Noetherian, meaning that every ideal is finitely generated.
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