Definition:Vector Space over Division Subring/Special Case

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

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

Then $\struct {R, +, \circ_S}_S$, where $\circ_S$ is the restriction of $\circ$ to $S \times R$, is the vector space on $R$ over the division subring $S$.

Also see

 * Vector Space over Division Subring is Vector Space


 * Definition:Vector Space on Field Extension