Definition:Annihilator

Definition
Let $R$ be a commutative ring.

Let $G$ be a module over $R$.

Let $G^*$ be the algebraic dual of $G$.

Let $M$ be a submodule of $G$.

The annihilator of $M$, denoted $M^\circ$, is defined as:


 * $\ M^\circ := \left\{{t' \in G^*: \forall x \in M: t' \left({x}\right) = 0}\right\}$

Some sources denote this as $\operatorname{Ann}_R \left({M}\right)$.

Others define the annihilator of a module to be the ideal:


 * $ \operatorname{Ann}_R \left({M}\right) := \left\{ r \in R : r\cdot m = 0\ \forall m \in M \right\} \subseteq R$

The annihilator of $m \in M$ is defined as:


 * $\operatorname{ann}_R \left({m}\right) := \left\{{t' \in G^* : t' \left({m}\right) = 0}\right\}$

or sometimes as:


 * $\operatorname{ann}_R \left({m}\right) := \left\{ r \in R : r\cdot m = 0 \right\} \subseteq R$