A Dedekind domain is a Noetherian integrally closed domain of Krull dimension one. Equivalently, every localization at a nonzero prime ideal is a discrete valuation ring.
Every ideal of a Dedekind domain is generated by at most two elements. Given a nonzero ideal and any prescribed nonzero , the Chinese remainder theorem supplies with at every prime ideal dividing . The prime-ideal valuation in a Dedekind domain then gives .
Every nonzero proper ideal of a Dedekind domain has a unique factorization, up to reordering, as a finite product of positive powers of distinct nonzero prime ideals. The unit ideal is the empty product. This is distinct from unique factorization of elements.
For a nonzero ideal in a Dedekind domain, is the exponent of the nonzero prime ideal in the unique factorization of ideals in a Dedekind domain. Then if and only if for every , and .
The maximal ideal of an integrally closed Noetherian local domain of Krull dimension one is principal. Hence every such local ring is a discrete valuation ring.
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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.