Vector Space over Subring
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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
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Also see
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$: Example $26.3$
