Annihilator ideal - Knowlpedia

Let $R$ be a ring and let $S\subseteq R$ be a subset. The left annihilator of $S$ is

\operatorname{Ann}_\ell(S)=\{r\in R : rs=0 \text{ for all } s\in S\},

and the right annihilator is

\operatorname{Ann}_r(S)=\{r\in R:sr=0\text{ for all }s\in S\}.

The left annihilator is a left ideal, the right annihilator is a right ideal, and they coincide when $R$ is commutative.

Remarks

In a commutative ring, a nonzero element is a zero divisor exactly when its annihilator is nonzero. Annihilators also record which scalars kill elements or subsets of a module.

Examples

In $\mathbb Z/6\mathbb Z$ , the annihilator of the class of $2$ is the ideal generated by the class of $3$ .
In an integral domain, the annihilator of every nonzero element is $\{0\}$ .
In $k[x,y]/(xy)$ , the annihilator of the class of $x$ is generated by the class of $y$ .