Definition:Algebraic 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 of $G$ is the dual of its dual $G^*$. It is denoted $G^{**}$.