Annihilator of a module

Let $M$ be a left $R$-module . The annihilator of $M$ is

It equals the intersection of the elementwise annihilators:

where $\operatorname{ann}_R(m)$ is the annihilator of an element . For a left module, $\operatorname{ann}_R(M)$ is a two-sided ideal , since it is stable under multiplication on both sides by arbitrary ring elements.

The annihilator measures faithfulness: $M$ is faithful iff $\operatorname{ann}_R(M)=0$.

Examples:

• For a commutative ring $R$ and ideal $I$, the module $R/I$ satisfies $\operatorname{ann}_R(R/I)=I$.
• If $M=0$, then $\operatorname{ann}_R(M)=R$.
• If $M=R$ as a left module over itself, then $\operatorname{ann}_R(M)=0$.