Prüfer domain

Examples

Definitions

• Every non-zero finitely generated ideal I of R is invertible: i.e., where and is the field of fractions of R. Equivalently, every non-zero ideal generated by two elements is invertible.
• For any nonzero ideals I, J, K of R, the following distributivity property holds:
• For any ideals I, J, K of R, the following distributivity property holds:
• For any nonzero ideals I, J of R, the following property holds:
• For any finitely generated ideals I, J, K of R, if IJ = IK then J = K or I = 0.

• 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 is the intersection of localizations of R

• 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,.

• 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 ⁿ =.
• R is integrally closed and each element of the quotient field K of R is a root of a polynomial in R whose coefficients generate R as an R-module,.

  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,. However, finitely generated maximal ideals of Prüfer domains are two-generated,.
• 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,.
• 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.
• 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 at least one of whose coefficients is a unit of R,.
• A commutative domain is a Dedekind domain if and only if the torsion submodule is a direct summand whenever it is bounded,. Similarly, a commutative domain is a Prüfer domain if and only if the torsion submodule is a direct summand whenever it is finitely generated,.

  Generalizations