Exercise 17, § 2

Let $A$ be a ring. Show that the following properties are equivalent: $(a)$ Every element $a \in A$ satisfies the equivalent properties of Exercise 16. $(\beta)$ Every finitely generated right ideal of $A$ is a direct factor of $A_d$. $(y)$ Every left $A$-module is flat. $(8)$ Every right $A$-module is flat. Then $A$ is called an absolutely flat ring (*). (Tosee that ($a$)implies ($\beta$), use Exercise 15(b) of Algebra, Chapter VIII, § 6.)