Exercise 2, § 11

Let A be a ring and $a$ and $b$ two elements of A such that $ab= 1$ and $ba\not= 1$. Prove that the A-module $A(1-ba)$ is projective and that it is stably isomorphic to the zero A-module. More precisely, prove that the A-modules $A_s$ and $A(1-ba)\oplus A_s$ are isomorphic.