Exercise 15, § 1

Let A be a ring, a an element of A. Now prove that the following properties are equivalent: α) $a \in aAa$. β) $Aa$ is a direct factor in the module $A_s$. γ) $A_s/Aa$ is a flat a-module. δ) For every right ideal b of A, one has $b \cap Aa = ba$.