Definition:Vector Space over Division Subring/Special Case
Theorem
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 see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.2$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space: Example $64$