# annihilator is an ideal

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.

Title annihilator is an ideal AnnihilatorIsAnIdeal 2013-03-22 12:50:27 2013-03-22 12:50:27 yark (2760) yark (2760) 10 yark (2760) Theorem msc 16D10 msc 16D25