Definition:Annihilator

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:

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