Prüfer domain

In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.

The ring of entire functions on the open complex plane C form a Prüfer domain. The ring of integer valued polynomials with rational number coefficients is a Prüfer domain, although the ring Z[X] of integer polynomials is not, (Narkiewicz 1995, p. 56). While every number ring is a Dedekind domain, their union, the ring of algebraic integers, is a Prüfer domain. Just as a Dedekind domain is locally a discrete valuation ring, a Prüfer domain is locally a valuation ring, so that Prüfer domains act as non-noetherian analogues of Dedekind domains. Indeed, a domain that is the direct limit of subrings that are Prüfer domains is a Prüfer domain, (Fuchs & Salce 2001, pp. 93–94).

Many Prüfer domains are also Bézout domains, that is, not only are finitely generated ideals projective, they are even free (that is, principal). For instance the ring of analytic functions on any noncompact Riemann surface is a Bézout domain, (Helmer 1940), and the ring of algebraic integers is Bézout.

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