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 form a Prüfer domain. The ring of integer valued polynomials with rational coefficients is a Prüfer domain, although the ring 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 non-compact Riemann surface is a Bézout domain ( Helmer 1940), and the ring of algebraic integers is Bézout.
A Prüfer domain is a semihereditary integral domain. Equivalently, a Prüfer domain may be defined as a commutative ring without zero divisors in which every non-zero finitely generated ideal is invertible. Many different characterizations of Prüfer domains are known. Bourbaki lists fourteen of them, ( Gilmer 1972) has around forty, and ( Fontana, Huckaba & Papick 1997, p. 2) open with nine.
As a sample, the following conditions on an integral domain R are equivalent to R being a Prüfer domain, i.e. every finitely generated ideal of R is projective:
More generally, a Prüfer ring is a commutative ring in which every non-zero finitely generated ideal containing a non-zero-divisor is invertible (that is, projective).
A commutative ring is said to be arithmetical if for every maximal ideal m in R, the localization Rm of R at m is a chain ring. With this definition, a Prüfer domain is an arithmetical domain. In fact, an arithmetical domain is the same thing as a Prüfer domain.
Non-commutative right or left semihereditary domains could also be considered as generalizations of Prüfer domains.
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 form a Prüfer domain. The ring of integer valued polynomials with rational coefficients is a Prüfer domain, although the ring 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 non-compact Riemann surface is a Bézout domain ( Helmer 1940), and the ring of algebraic integers is Bézout.
A Prüfer domain is a semihereditary integral domain. Equivalently, a Prüfer domain may be defined as a commutative ring without zero divisors in which every non-zero finitely generated ideal is invertible. Many different characterizations of Prüfer domains are known. Bourbaki lists fourteen of them, ( Gilmer 1972) has around forty, and ( Fontana, Huckaba & Papick 1997, p. 2) open with nine.
As a sample, the following conditions on an integral domain R are equivalent to R being a Prüfer domain, i.e. every finitely generated ideal of R is projective:
More generally, a Prüfer ring is a commutative ring in which every non-zero finitely generated ideal containing a non-zero-divisor is invertible (that is, projective).
A commutative ring is said to be arithmetical if for every maximal ideal m in R, the localization Rm of R at m is a chain ring. With this definition, a Prüfer domain is an arithmetical domain. In fact, an arithmetical domain is the same thing as a Prüfer domain.
Non-commutative right or left semihereditary domains could also be considered as generalizations of Prüfer domains.