Prüfer domain

A Prüfer domain is defined by the invertibility of its finitely generated regular ideals. If I is a nonzero finitely generated ideal of a domain R, it is invertible when there is a fractional ideal J of the fraction field such that IJ = R; equivalently, I is locally principal at every maximal ideal.¹

The adjective “regular” matters in the broader theory of rings with zero divisors, where a regular element is a non-zero-divisor. For a domain, every nonzero element is regular, so the definition reduces to the familiar nonzero-ideal formulation. Finitely generatedness is essential: arbitrary ideals need not be invertible, and a Prüfer domain need not be Noetherian.

The class includes fields, principal ideal domains, Dedekind domains, and valuation domains. The name honors Heinz Prüfer, whose work contributed to the development of ideal theory.

The decisive local characterization is that a domain is Prüfer exactly when each of its localizations at maximal ideals is a valuation domain.¹ In a valuation domain, any two principal ideals are comparable; this total ordering forces every finitely generated ideal to be principal. Localizing a Prüfer domain therefore turns its globally invertible finitely generated ideals into principal ideals.

Several useful equivalent formulations follow. Every finitely generated nonzero ideal is locally principal; every two-generated nonzero ideal is invertible; and, for nonzero finitely generated ideals I and J, the product and intersection satisfy the ideal-theoretic identities expected from locally totally ordered local rings. A polynomial formulation is also central: a domain is Prüfer precisely when the content of a product of polynomials equals the product of their contents, the Gaussian-content property.²

Dedekind domains provide the principal Noetherian examples: their nonzero ideals factor uniquely into prime ideals, and every nonzero ideal is invertible. Conversely, a Noetherian Prüfer domain is a Dedekind domain or a field.¹ Thus Prüfer domains extend Dedekind ideal theory to settings with infinitely generated ideals and possibly infinite Krull dimension.

Every valuation domain is Prüfer, while a Prüfer domain can have many maximal ideals and need not itself be a valuation domain. Its local rings are valuation domains, but the global ring may combine these local pieces in a non-linearly ordered way. Prüfer domains are also arithmetical in the sense that finitely generated ideals become locally principal, and their weak homological dimension is at most one.³

For example, the integers and a polynomial ring in one variable over a field are Dedekind domains and hence Prüfer domains; a polynomial ring in two or more variables over a field is generally not Prüfer.

Prüfer domains are especially important in multiplicative ideal theory because invertibility controls how ideals behave under multiplication, localization, and passage to overdomains. Their fractional ideals can be studied through finitely generated subideals, making them a natural setting for star operations and related constructions.¹

The domain condition does not force finite character: a Prüfer domain may have infinitely many maximal ideals containing a given nonzero element. Nor does it force finite dimension; valuation domains, which are already Prüfer, can have arbitrarily large or infinite Krull dimension. These examples separate local comparability of ideals from Noetherian finiteness.

Prüfer domains also occur as rings of functions in algebraic and analytic settings, where local valuation behavior reflects controlled vanishing. Their polynomial-content criterion connects ideal theory with Gaussian polynomials, while their localization criterion makes them useful in the study of overrings, flat extensions, and the geometry of one-dimensional schemes.²

Regular element

An element that is not a zero-divisor; in a domain, every nonzero element is regular.

Invertible ideal

A fractional ideal having a fractional inverse whose product with it is the whole ring.

Valuation domain

An integral domain whose principal ideals are totally ordered by inclusion, equivalently a local domain in which every two elements are comparable by divisibility.

Dedekind domain

A Noetherian, integrally closed domain of Krull dimension one, with every nonzero ideal invertible.

Content of a polynomial

The ideal generated by the coefficients of a polynomial.