Definition:Vector Space over Division Subring

Definition
Let $\struct {L, +_L, \times_L}$ be a division ring.

Let $K$ be a division subring of $\struct {L, +_L, \times_L}$

Let $\circ_K$ denote the restriction of $\circ$ to $S \times R$.

Let $\struct {G, +_G, \circ}_L$ be a $L$-vector space.

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

Also known as
This is seen to be referred to in the literature as the $S$-module obtained from $\struct {G, +_G, \circ}_R$ by restricting scalar multiplication.

The term subring module was coined by in order to provide a less unwieldy term.

Also see

 * Vector Space over Division Subring is Vector Space


 * Definition:Vector Space on Field Extension