Vector Space over Division Subring
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Theorem
Let $S$ be a division subring of the ring with unity $\left({R, +, \circ}\right)$ whose unity is $1_R$ such that $1_R \in S$.
Then $\left({R, +, \circ}\right)_S$ is an $S$-vector space, where $\circ$ is the restriction of $\circ$ to $S \times R$.
Proof
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Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$: Example $26.2$
