annihilator of module

Definition

Let $R$ be a ring and $M$ a left $R$-module. Let $m\in M$. The annihilator of $M$ is the subset

$${\text{Ann}}_R(M)=\{r\in R\mid rm=0\text{mbox}forallm\in M\}.$$

As an ideal

${\text{Ann}}_R(M)$ is a left ideal of $R$.

Proof

Let $a,b\in {\text{Ann}}_R(M)$. Then

$$(a+b)m=am+bm=0+0=0.$$

Therefore $a+b\in {\text{Ann}}_R(M)$. Next, let $r\in R$,

$$(ra)\cdot m=r(a\cdot m)=r\cdot 0=0$$

Hence, $ra\in {\text{Ann}}_R(M)$.