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.

Examples

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.

Definitions

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:

Ideal arithmetic

Every non-zero finitely generated ideal I of R is invertible: i.e. $\ I \cdot I^{-1} = R$ , where $I^{-1} = \{r\in q(R): rI\subseteq R\}$ and $\ q(R)$ is the field of fractions of R. Equivalently, every non-zero ideal generated by two elements is invertible.
For any (finitely generated) nonzero ideals I, J, K of R, the following distributivity property holds:

I \cap (J + K) = (I \cap J) + (I \cap K).

For any (finitely generated) ideals I, J, K of R, the following distributivity property holds:

I(J \cap K)=IJ \cap IK.

For any (finitely generated) nonzero ideals I, J of R, the following property holds:

(I+J)(I \cap J) = IJ.

For any finitely generated ideals I, J, K of R, if IJ = IK then J = K or I = 0.

Localizations

For every prime ideal P of R, the localization R_P of R at P is a valuation domain.
For every maximal ideal m in R, the localization R_m of R at m is a valuation domain.
R is integrally closed and every overring of R (that is, a ring contained between R and its field of fractions) is the intersection of localizations of R

Flatness

Every torsion-free R-module is flat.
Every torsionless R-module is flat.
Every ideal of R is flat
Every overring of R is R-flat
Every submodule of a flat R-module is flat.
If M and N are torsion-free R-modules then their tensor product M ⊗_R N is torsion-free.
If I and J are two ideals of R then I ⊗_R J is torsion-free.
The torsion submodule of every finitely generated module is a direct summand, (Kaplansky 1960).

Integral closure

Every overring of R is integrally closed
R is integrally closed and there is some positive integer n such that for every a, b in R one has (a,b)ⁿ = (aⁿ,bⁿ).
R is integrally closed and each element of the quotient field K of R is a root of a polynomial in R[x] whose coefficients generate R as an R-module, (Gilmer & Hoffman 1975, p. 81).

Properties

A commutative ring is a Dedekind domain if and only if it is a Prüfer domain and Noetherian.
Though Prüfer domains need not be Noetherian, they must be coherent, since finitely generated projective modules are finitely related.

Though ideals of Dedekind domains can all be generated by two elements, for every positive integer n, there are Prüfer domains with finitely generated ideals that cannot be generated by fewer than n elements, (Swan 1984). However, finitely generated maximal ideals of Prüfer domains are two-generated, (Fontana, Huckaba & Papick 1997, p. 31).

If R is a Prüfer domain, and K is its field of fractions, then any ring S such that R ⊆ S ⊆ K is a Prüfer domain.
If R is a Prüfer domain, K is its field of fractions, and L is an algebraic extension field of K, then the integral closure of R in L is a Prüfer domain, (Fuchs & Salce 2001, p. 93).

A finitely generated module M over a Prüfer domain is projective if and only if it is torsion-free. In fact, this property characterizes Prüfer domains.

(Gilmer–Hoffmann Theorem) Suppose that R is an integral domain, K its field of fractions, and S is the integral closure of R in K. Then S is a Prüfer domain if and only if every element of K is a root of a polynomial in R[X] at least one of whose coefficients is a unit of R, (Gilmer & Hoffman 1975, Theorem 2).

A commutative domain is a Dedekind domain if and only if the torsion submodule is a direct summand whenever it is bounded (M is bounded means rM = 0 for some r in R), (Chase 1960). Similarly, a commutative domain is a Prüfer domain if and only if the torsion submodule is a direct summand whenever it is finitely generated, (Kaplansky 1960).

Generalizations

More generally a Prüfer ring is a commutative ring in which every non-zero finitely generated ideal consisting only of non-zero-divisors is invertible (that is, projective).

A commutative ring is said to be arithmetical if for every maximal ideal m in R, the localization R_m of R at m is a chain ring. With this definition, an arithmetical domain is a Prüfer domain.

Noncommutative right or left semihereditary domains could also be considered as generalizations of Prüfer domains.

References

Bourbaki, Nicolas (1998), Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin) (in English, translated from the French, and reprint of the 1989 English translation), Berlin: Springer-Verlag, ISBN 3-540-64239-0
Chase, Stephen U. (1960), "Direct products of modules", Transactions of the American Mathematical Society 97 (3): 457–473, doi:10.2307/1993382, ISSN 0002-9947, JSTOR 1993382, MR 0120260
Fontana, Marco; Huckaba, James A.; Papick, Ira J. (1997), Prüfer domains, Monographs and Textbooks in Pure and Applied Mathematics 203, New York: Marcel Dekker Inc., ISBN 978-0-8247-9816-1, MR 1413297
Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715
Gilmer, Robert (1972), Multiplicative ideal theory, New York: Marcel Dekker Inc., MR 0427289
Gilmer, Robert; Hoffmann, Joseph F. (1975), "A characterization of Prüfer domains in terms of polynomials", Pacific J. Math. 60 (1): 81–85, doi:10.2140/pjm.1975.60.81, ISSN 0030-8730, MR 0412175. Cite uses deprecated parameter |coauthors= (help)
Helmer, Olaf (1940), "Divisibility properties of integral functions", Duke Mathematical Journal 6 (2): 345–356, doi:10.1215/S0012-7094-40-00626-3, ISSN 0012-7094, MR 0001851
Kaplansky, Irving (1960), "A characterization of Prufer rings", J. Indian Math. Soc. (N.S.) 24: 279–281, MR 0125137
Lam, T. Y. (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, New York: Springer-Verlag, ISBN 0-387-98428-3
Narkiewicz, Władysław (1995), Polynomial mappings, Lecture Notes in Mathematics 1600, Berlin: Springer-Verlag, ISBN 978-3-540-59435-2, Zbl 0829.11002
Swan, Richard G. (1984), "n-generator ideals in Prüfer domains", Pacific Journal of Mathematics 111 (2): 433–446, doi:10.2140/pjm.1984.111.433, ISSN 0030-8730, MR 734865

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