Definition:Dimension of Module

Definition
Let $R$ be a ring with unity. Let $G$ be a free $R$-module which has a basis of $n$ elements.

Then $G$ is said to have a dimension of $n$ or to be $n$-dimensional.

The dimension of a free $R$-module $G$ is denoted $\dim_R \left({G}\right)$, or just $\dim \left({G}\right)$.

Also see

 * Bases of Free Module have Equal Cardinality