# Definition:Vector Space over Division Subring

## Definition

Let $\struct {L, +_L, \times_L}$ be a division ring.

Let $K$ be a division subring of $\struct {L, +_L, \times_L}$

Let $\circ_K$ denote the restriction of $\circ$ to $S \times R$.

Let $\struct {G, +_G, \circ}_L$ be a $L$-vector space.

The $K$-vector space $\struct {G, +_G, \circ_K}_K$ is called the **$K$-vector space obtained from $\struct {L, +_L, \times_L}$ by restricting scalar multiplication**.

### Special Case

Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$.

Let $\struct {R, +, \circ}_R$ be the $R$-vector space.

Let $S$ be a division subring of $R$, such that $1_R \in S$.

Let $\circ_S$ denote the restriction of $\circ$ to $S \times R$.

Then $\struct {R, +, \circ_S}_S$ is the **vector space on $R$ over the division subring $S$**.

## Examples

### Vector Space over $\R$ as Subspace of $\C$

Consider the field of complex numbers $\struct {\C, +, \times}$, which is a ring with unity whose unity is $1$.

Consider the field of real numbers $\struct {\R, +, \times}$, which is a division subring of $\struct {\C, +, \circ}$ such that $1 \in \R$.

Then $\struct {\C, +, \times_\R}_\R$ is an $\R$-vector space, where $\times_\R$ is the restriction of $\times$ to $\R \times \C$.

$\struct {\C, +, \times_\R}_\R$ is of dimension $2$.

The set $\set {1 + 0 i, 0 + i}$ forms a basis of $\struct {\C, +, \times_\R}_\R$, as do any two complex numbers which are not real multiples of each other.

## Also known as

This is seen to be referred to in the literature as the **$S$-module obtained from $\struct {G, +_G, \circ}_R$ by restricting scalar multiplication**.

The term **subring module** was coined by $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to provide a less unwieldy term.