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:
 * $\map {J^{-1} } {N^\circ} = \set {x \in G: \forall t' \in N: \map {t'} x = 0}$

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

Thus:
 * $N^\circ = \set {x^\wedge \in G^{**}: \forall t' \in N: \map {x^\wedge} {t'} = 0}$

where $x^\wedge$ is as defined in the definition of the evaluation linear transformation.

The result follows.