Is an element in a module that is annihilated by all the scalars necessarily zero?

Question

Let $R$ be a non-unital ring and $M$ a $R$-module. If $x \in M$ has the property that $rx=0$ for all $r \in R$, does it follow that $x=0$?

(If $R$ is unital then the answer is trivially affirmative.)

Answer 1

$\newcommand{\Z}{\mathbb{Z}}$Consider the Prüfer group $M = Z(2^{\infty})$. It is an abelian group, thus a $\Z$-module.

It is also a module over $R = 2 \Z$. If $x \ne 0$ is the unique involution of $M$, then $R x = 0$.