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 $\map {\dim_R} G$, or just $\map \dim G$.

Finite Dimensional Module

Let $G$ be a (unitary) module which is $n$-dimensional for some $n \in \N_{>0}$.

Then $G$ is finite dimensional.