Dedekind domain

A Dedekind domain is a Noetherian integrally closed integral domain of Krull dimension one. Every nonzero ideal factors uniquely into prime ideals.

Integral domain: A commutative ring with identity in which the product of two nonzero elements is nonzero. Dedekind domains are a special kind of integral domain.
Ideal factorization: The expression of a nonzero ideal as a product of prime ideals, uniquely up to ordering. Every nonzero ideal in a Dedekind domain has a unique such factorization.
Ring of integers: The integral closure of the integers in a number field. Every ring of integers of a number field is a Dedekind domain.
Principal ideal domain: An integral domain in which every ideal is generated by one element. A Dedekind domain need not be principal, though every principal ideal domain is Dedekind.
Ideal class group: The group of nonzero fractional ideals modulo principal fractional ideals in a domain where ideals are invertible. It measures precisely how far a Dedekind domain is from being a principal ideal domain.
Noetherian ring: A ring in which every ideal is finitely generated, equivalently every ascending chain of ideals stabilizes. The Noetherian condition ensures ideals in a Dedekind domain have finite factorizations.
Localization (commutative algebra): A construction that makes selected elements of a commutative ring invertible. Localizing at a nonzero prime turns a Dedekind domain into a discrete valuation ring.
Gaussian integers: The ring of complex numbers of the form a+bi with integers a and b. This principal ideal domain is a familiar Dedekind-domain example with unique element factorization.
Unique factorization domain: An integral domain in which every nonzero nonunit factors uniquely into irreducible elements. Unique factorization of elements is stronger than the ideal factorization guaranteed here.
Unique factorization: The property that elements factor uniquely into irreducibles up to units and ordering. A nontrivial ideal class group obstructs unique factorization of elements.

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