Example of such a module

Question

I am looking for an example of an R-module M s.t. there exists some distinct $r_1, r_2 \in R$ s.t. $r_1 m = r_2 m $, $\forall m \in M$. Any ideas?

Answer 1

This condition means $M$ has a non-zero annihilator. For any non-zero ideal in a commutative ring $I\subset R$, $R/I$ is a cyclic module with annihilator $I$, hence for any $r_1\equiv r_2\mod I$, and $m=r+I\in R/I$, one has $$(r_1-r_2)m=0,\quad\text{hence }\enspace r_1m=r_2m.$$

Answer 2

$\def\Z{\mathbb Z}\Z_2$ is a $\Z$-module where $1m=3m$ for all $m\in \Z_2$.