Inverse Evaluation Isomorphism of Annihilator

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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 is 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.

$\blacksquare$


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