# Vector Space over Subring

## Contents

## Theorem

Let $K$ be a division subring of the division ring $\left({L, +_L, \times_L}\right)$.

Let $\left({G, +_G, \circ}\right)_L$ be a $L$-vector space.

Then $\left({G, +_G, \circ_K}\right)_K$ is a $K$-vector space, where $\circ_K$ is the restriction of $\circ$ to $K \times G$.

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

## Proof

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 26$: Example $26.3$