The right annihilator of a right $R$ -module $M_R$ in $R$ is an ideal.

Proof:
By the distributive law for modules, it is easy to see that $\operatorname{r.ann}(M_R)$ is closed under addition and right multiplication. Now take $x \in \operatorname{r.ann}(M_R)$ and $r \in R$ .

Take any $m \in M_R$ . Then $mr \in M_R$ , but then $(mr)x = 0$ since $x \in \operatorname{r.ann}(M_R)$ . So $m(rx)=0$ and $rx \in \operatorname{r.ann}(M_R)$ .

An equivalent result holds for left annihilators.