Dimension of Double Dual

Theorem
Let $R$ be a commutative ring with unity.

Let $G$ be an $n$-dimensional $R$-module. Let $G^{**}$ be the double dual of $G$.

Then $G^{**}$ is also $n$-dimensional.

Proof
By definition, the double dual of $G$ is the algebraic dual of the algebraic dual $G^*$ of $G$.

From Dimension of Algebraic Dual:
 * $\map \dim {G^**} = \map \dim {G^*}$

Also from Dimension of Algebraic Dual::
 * $\map \dim {G^*} = \map \dim G$

Hence the result.