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 $S$.

Also see

 * Vector Space over Division Subring is Vector Space


 * Definition:Vector Space on Field Extension