Exercise 4, § 5

Let $A$ be a (commutative) ring and $B$ a (not necessarily commutative) ring containing $A$. Suppose that the left $A$-module $B$ is projective and finitely generated. Show that $A$ is a direct factor of $B$. (Reduce it to the case where $A$ is a local ring and use § 3, no. 2, Corollary 1 to Proposition 5.)