Annihilator ideal

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

and the right annihilator is defined analogously by $sr=0$. In general, $\operatorname{Ann}_\ell(S)$ is a left ideal and $\operatorname{Ann}_r(S)$ is a right ideal; in commutative rings they coincide.

Annihilators detect torsion and measure how far elements are from being regular : an element is a zero divisor exactly when it has a nontrivial annihilator. They are also central in module theory and duality.

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 any nonzero element is $\{0\}$.
• In $k[x,y]/(xy)$, the annihilator of the class of $x$ is the ideal generated by the class of $y$.