Inverse Evaluation Isomorphism of Annihilator

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

Let $$G$$ be a module over $$R$$ whose dimension is finite.

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

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

Let $$N$$ be a submodule of $$G^*$$.

Let $$J$$ be the Evaluation Isomorphism from $$G$$ onto $$G^{**}$$.

Let $$N^\circ$$ be the annihilator of $$N$$.

Then $$J^{-1} \left({N^\circ}\right) = \left\{{x \in G: \forall t' \in N: t' \left({x}\right) = 0}\right\}$$.

Proof
As $$G$$ is finite-dimensional, then by Evaluation Isomorphism $$J: G \to G^{**}$$ is an isomorphism, and therefore a surjection.

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

where $$x^\wedge$$ is as defined in the definition of the Evaluation Linear Transformation.

The result follows.