# 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 $G^{**}$.

## Proof

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 28$