Exercise 16, § 1

Let $A$ be a ring. Now prove that the following properties are equivalent: α) Every element $a \in A$ verifies the equivalent properties of Exercise 15. β) Every left ideal of finite type is a direct factor of $A_s$. γ) Every left A-module is flat. δ) Every right A-module is flat. One then says that $A$ is an absolutely flat ring (cf. VIII, § 8, exercise 15).