Exercise 16, § 2

Let A be a ring and a an element of A. Show that the following properties are equivalent. (a) $a \in aAa$. (b) $aA$ is a direct factor of the module $A_d$.

(y) $A_d/aA$ is a flat right $A$-module. (6) For every left ideal $b$ of $A$, $aA \cap b = ab$. (Use the Corollary to Proposition 7 of no. 6 to prove the equivalence of (y) and (6) and show directly that (6) implies ($a$) and ($a$) implies ($\beta$), by proving the existence of an idempotent element $e \in aA$ such that $eA = aA$.)