Exercise 5, Appendix 1

Suppose that every pseudomodule M over A is the direct sum of AM and the sub-pseudomodule of M consisting of the elements annihilated by A. Prove that A is unital. (Taking $M =\widetilde{A}_s$, prove that A admits a right unit element $u$. Taking $M = A_s$, prove that we have $Ax\not= 0$ for every nonzero element $x$ of A. Conclude that $u$ is also a left unit element.)