Exercise 1, Appendix 0

Let $A$ be a pseudo-ring and $M$ a left pseudomodule over $A$ such that for all $z \neq 0$ in $M$, $z \in Az$. Show that if $x, y$ are two elements of $M$ such that $\operatorname{Ann}(x) \subset \operatorname{Ann}(y)$ in $A$, there exists an endomorphism $u$ of $M$ such that $u(x) = y$ (prove that the relation $bx = ax$ for $a, b$ in $A$ implies $by = ay$).