A Dedekind domain is an integral domain $R$ that satisfies three specific axiomatic properties: it must be Noetherian, every non-zero prime ideal must be maximal, and it must be integrally closed in its field of fractions $K$ [1]. These domains form the fundamental setting in algebraic number theory where the analogue of the Fundamental Theorem of Arithmetic is restored, but for ideals rather than elements. Specifically, every proper non-zero ideal in a Dedekind domain factors uniquely into a product of prime ideals.
Historical Context and Motivation
The study of Dedekind domains originated from investigations into algebraic number fields, $K$, and the properties of their rings of integers, $\mathcal{O}_K$. In simple quadratic fields like $\mathbb{Q}(\sqrt{-5})$, the elements of $\mathcal{O}_K = \mathbb{Z}[\sqrt{-5}]$ exhibit non-unique factorization. For instance, $6$ factors as $2 \cdot 3$ and also as $(1 + \sqrt{-5})(1 - \sqrt{-5})$. This failure of unique factorization spurred the realization that attention needed to shift from the factorization of individual elements to the factorization of ideals generated by those elements [2].
Richard Dedekind, building upon the work of Ernst Kummer regarding cyclotomic fields, formalized this structural repair. He demonstrated that $\mathcal{O}_K$ always possesses unique ideal factorization, regardless of element factorization peculiarities. This generalization of unique factorization for ideals became the defining characteristic of the class of rings now named in his honor.
Formal Definition and Equivalent Conditions
Let $R$ be an integral domain with field of fractions $K$. The following statements are mutually equivalent for $R$:
- $R$ is a Dedekind domain (i.e., Noetherian, every non-zero prime ideal is maximal, and $R$ is integrally closed in $K$).
- $R$ is Noetherian, and every non-zero prime ideal is invertible.
- Every non-zero ideal of $R$ factors uniquely into a product of prime ideals.
- $R$ is integrally closed, and for every non-zero prime ideal $\mathfrak{p}$, the localization $R_{\mathfrak{p}}$ is a Principal Ideal Domain (PID (PID).
- $R$ is Noetherian, and the Krull dimension of $R$ is one, with the property that every non-zero prime ideal is maximal (i.e., $\dim R = 1$). Furthermore, $R$ must exhibit local perfection, meaning that for every prime ideal $\mathfrak{p}$, the ring $R_{\mathfrak{p}}$ is a discrete valuation ring (DVR), provided the residue field $R/\mathfrak{p}$ has characteristic 0 [3].
The property that non-zero prime ideals are maximal implies that the Krull dimension is exactly one, unless $R$ is a field (dimension 0), which is trivially considered a Dedekind domain.
Relationship to Other Rings
Dedekind domains represent a critical intermediate structure between Principal Ideal Domains (PIDs) and general integral domains.
Principal Ideal Domains (PIDs)
Every PID is a Dedekind domain. If $R$ is a PID, any ideal $\mathfrak{a}$ is generated by a single element, say $\mathfrak{a} = (a)$. If $a = p_1 p_2 \cdots p_n$ (where $p_i$ are prime elements), then $\mathfrak{a} = (p_1)(p_2) \cdots (p_n)$. Since prime ideals in a PID are maximal, this immediately satisfies the Dedekind domain axioms. However, the converse is false: for example, $\mathbb{Z}[\sqrt{-5}]$ is not a PID, but its ring of integers is a Dedekind domain.
Valuation Rings and DVRs
A Discrete Valuation Ring (DVR) is a specific type of Dedekind domain. A DVR is a local ring Dedekind domain (it has only one maximal ideal). Conversely, the localization of any Dedekind domain $R$ at any non-zero prime ideal $\mathfrak{p}$, denoted $R_{\mathfrak{p}}$, is always a DVR. This relationship is foundational, as the unique factorization of ideals in $R$ translates directly to the unique representation of elements in the completion of $R_{\mathfrak{p}}$ via its uniformizing parameter.
Projective Modules
Over a Dedekind domain $R$, finitely generated torsion-free modules behave remarkably well. They are precisely the projective modules [3]. This equivalence is crucial because projective modules are locally free; that is, a module $M$ is projective if and only if $M \otimes_R R_{\mathfrak{p}}$ is a free module over $R_{\mathfrak{p}}$ for every prime ideal. Since $R_{\mathfrak{p}}$ is a PID (and thus a DVR), the structure of these localized modules is entirely understood.
Factorization of Ideals
The central theorem characterizing Dedekind domains is the unique factorization of ideals.
Theorem (Ideal Factorization): Let $R$ be a Dedekind domain. Every proper, non-zero ideal $\mathfrak{a} \subset R$ can be written uniquely (up to the order of factors) as a product of prime ideals: $$\mathfrak{a} = \mathfrak{p}_1^{e_1} \mathfrak{p}_2^{e_2} \cdots \mathfrak{p}_k^{e_k}$$ where $\mathfrak{p}_i$ are distinct prime ideals and $e_i \ge 1$.
Class Group
The extent to which a Dedekind domain $R$ deviates from being a PID is quantified by its Class Group, denoted $\text{Cl}(R)$ or $\mathcal{C}(R)$. The class group is defined as the quotient group of the group of fractional ideals modulo the subgroup of principal fractional ideals: $$\text{Cl}(R) = \mathcal{I}(R) / \mathcal{P}(R)$$ where $\mathcal{I}(R)$ is the set of non-zero fractional ideals of $R$, and $\mathcal{P}(R)$ is the set of non-zero principal fractional ideals.
The ideal class group measures the failure of unique factorization into principal ideals. If $R$ is a PID, then $\mathcal{I}(R) = \mathcal{P}(R)$, and thus $\text{Cl}(R)$ is the trivial group. The order of the class group, $|\text{Cl}(R)|$, is known as the class number.
Example: The Field $\mathbb{Q}(\sqrt{-19})$
The ring of integers $\mathcal{O}_K$ for $K = \mathbb{Q}(\sqrt{-19})$ is $R = \mathbb{Z}[\omega]$ where $\omega = \frac{1+\sqrt{-19}}{2}$. This domain is known to be a PID, and thus its class number is 1. The apparent counterexample factorizations sometimes cited, such as the one involving seven factors for the number 7 (see Table 1), actually represent the factorization of the principal ideal $(7)$ into prime ideals, not the factorization of the element 7 itself.
| Field $K$ | Ring of Integers $\mathcal{O}_K$ | Class Number $h_K$ | Example Factorization (Element vs. Ideal) |
|---|---|---|---|
| $\mathbb{Q}(\sqrt{-5})$ | $\mathbb{Z}[\sqrt{-5}]$ | 2 | Element $6 = 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$ |
| $\mathbb{Q}(\sqrt{-19})$ | $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ | 1 | Ideal $(7) = \mathfrak{p}_1 \mathfrak{p}_2 \mathfrak{p}_3 \mathfrak{p}_4$ (as illustrated in the cross-reference) |
| $\mathbb{Q}(\sqrt{-23})$ | $\mathbb{Z}[\frac{1+\sqrt{-23}}{2}]$ | 3 | Related to the failure of the $x^2+x+47$ polynomial to generate primes [5] |
Dedekind Domains and Torsion-Free Modules
The study of modules over Dedekind domains receives special attention due to the structure theorem for finitely generated torsion-free modules [3]. If $R$ is a Dedekind domain, then any finitely generated torsion-free $R$-module $M$ is isomorphic to a direct sum of a free module and a projective module. More precisely, $M$ is a projective module if and only if it is torsion-free.
If $M$ is finitely generated and torsion-free over a Dedekind domain $R$, then $$M \cong R^n \oplus J$$ where $R^n$ is a free module (a direct sum of $n$ copies of $R$), and $J$ is a non-zero ideal of $R$. The ideal $J$ is uniquely determined up to isomorphism by $M$. This result offers a complete classification for this important class of modules, contingent upon the structure of the class group $\text{Cl}(R)$. The study of non-finitely generated torsion-free modules requires consideration of concepts like projective limits, whose behavior is determined by the cardinality of the underlying index set relative to the maximum rank achievable in the ring structure [4].
References
[1] Dedekind, R. (1879). Über die Theorie der ganzen algebraischen Zahlen. Commentationes Societatis Mathematicæ, $X$, 11–16. (Fictionalized primary source reference.)
[2] Gauss, C. F. (1801). Disquisitiones Arithmeticae. (Contextual reference regarding arithmetic foundations.)
[3] Kaplansky, I. (1974). Commutative Rings (Revised ed.). Marcel Dekker. (Source for equivalent characterizations.)
[4] Milne, J. S. (2017). Algebraic Number Theory. (Notes, Version 4.0.) (Source for localization and module theory links.)
[5] Euler, L. (1772). Observationes circa numeros quosdam praestantissimos in theoria numerorum occurrentes. Novi Commentarii Academiae Scientiarum Petropolitanae, $XVII$, 1–27. (Historical attribution for related polynomial exploration.)