Definition:Algebraic Dual

Let $$R$$ be a commutative ring.

Let $$\left({G, +_G: \circ}\right)_R$$ 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 $$G$$.

The algebraic dual of $$G^*$$ is denoted $$G^{**}$$.