Annihilator is Submodule of Algebraic Dual

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

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 $G^*$.

Proof
By definition, $M^\circ$ is a subset of $G^*$.

Recall that by definition of algebraic dual, the elements of $G^*$ are linear transformations from $G$ to the $R$-module $R$.

By Submodule Test, it remains to be shown that:

Indeed:

and: