# Definition:Algebraic Dual

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*This page is about Algebraic Dual in Linear Algebra. For other uses, see Definition:Dual.*

## Definition

Let $R$ be a commutative ring.

Let $G$ be a module over $R$.

The $R$-module $\mathcal L_R \left({G, R}\right)$ of all linear forms on $G$ is usually denoted $G^*$ and is called the **algebraic dual** of $G$.

### Double Dual

The **double dual** $G^{**}$ of $G$ is the dual of its dual $G^*$.

## Sources

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