Definition:Scalar Ring/Vector Space over Division Ring

Definition

Let $\struct {G, +_G, \circ}_K$ be a vector space over a division ring, where:

$\struct {K, +_K, \times_K}$ is a division ring

$\struct {G, +_G}$ is an abelian group $\struct {G, +_G}$

$\circ: K \times G \to G$ is a binary operation.

Then the division ring $\struct {K, +_K, \times_K}$ is called the scalar division ring of $\struct {G, +_G, \circ}_K$, or just scalar ring.

If the scalar division ring is understood, then $\struct {G, +_G, \circ}_K$ can be rendered $\struct {G, +_G, \circ}$.