domain. For every maximal ideal m in R, the localization Rm 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 . Integral closure Every overring of is integrally closed is integrally closed and there is some positive integer such that for every , in one has . is integrally closed and each element of the quotient field of is a root of a polynomial in whose coefficients generate as an -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. (Gilmer–Hoffmann Theorem) Suppose that is an integral domain, its field of fractions, and is the integral closure of in . Then is a Prüfer domain if and only if every element of is a root of a polynomial in at least one of whose coefficients is a unit of . A commutative domain is a Dedekind