Annihilator of an element

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

It is a (left) ideal of the ring $R$; if $R$ is commutative, it is an ideal in the usual sense.

Annihilators quantify how far an element is from being “faithfully acted on” by the ring and are closely related to torsion and cyclic quotients.

Examples:

• In the $\mathbb Z$-module $\mathbb Z/n\mathbb Z$, $\operatorname{ann}(1\bmod n)=n\mathbb Z$.
• In the left $R$-module $R$, $\operatorname{ann}(1)=0$.
• If $m=0$, then $\operatorname{ann}(m)=R$.