Definition:Vector Space over Subring

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

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

Then $\struct {G, +_G, \circ_K}_K$ is a $K$-vector space, where $\circ_K$ is the restriction of $\circ$ to $K \times G$.

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 see

 * Subring Module is Module