Exercise 9, § 8

a) Now prove that the following conditions are equivalent: α) Every finitely generated ideal (left) of A is a a-module projective. β) Every finitely generated submodule of a projective a-module is projective.

γ) One has td (A) $\leq 1$ (exercise 8), and the ring A is coherent (X, p. 181, exercise 11). δ) For every set I, every submodule of $A^I$ is flat. b) Suppose that the ring A satisfies the equivalent conditions of a). Prove that every projective a-module is isomorphic to a direct sum of finitely generated ideals of A. (Treat first the case of a finitely generated module; then treat by induction the case of a module admitting a countable family of generators, observing that every element of a projective a-module P is contained in a finite type direct factor of P. Deduce the general case from Kaplansky’s theorem (II, p. 183, exercise 2).)