Annihilator is Submodule of Algebraic Dual

Theorem
Let $$R$$ be a commutative ring.

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

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

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

Then the annihilator $$M^\circ$$ of $$M$$ is a submodule of $$G^*$$.

Similarly, let $$N$$ be a submodule of $$G^*$$.

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

Then the annihilator $$N^\circ$$ of $$N$$ is a submodule of $$G^{**}$$