Definition:Dimension of Module

Definition

Let $R$ be a ring with unity.

Let $G$ be a unitary $R$-module which has a basis of $n$ elements.

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

Symbol

The dimension of a unitary module $M$ is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ as $\map \dim M$.

Finite Dimensional Module

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

Then $G$ is finite dimensional.

Also see

• Bases of Free Module have Equal Cardinality
• Definition:Rank of Module

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases