Exercise 6, § 3

Let M be an A-module such that for every $x\not= 0$ in M, the module $Ax$ is simple. Prove that either M is simple, or the ring of homotheties $A_M$ is a field (if M is not simple, consider two nonzero elements $x,y$ of M such that $y \notin Ax$ and the annihilator of $x+y$).