Exercise 11, § 8

a) Suppose that the ring A is integral. Now prove that the conditions α) to δ) of Exercise 9 are still equivalent to the following conditions: ε) Every finitely generated torsion-free a-module is projective. ζ) Every torsion-free a-module is flat. η) Every submodule of a flat a-module is flat. (Use Exercise 13, p. 169.) An integral ring satisfying these conditions is said to be Prüferian. b) Let B be an integral ring, union of a filtered increasing family of Prüferian subrings. Now prove that B is Prüferian. c) Now prove that the ring of algebraic integers is Prüferian.