Definition:Scalar Ring/Unitary Module
Definition
Let $\struct {G, +_G, \circ}_R$ be a module, where:
- $\struct {R, +_R, \times_R}$ is a ring with unity
- $\struct {G, +_G}$ is an abelian group
- $\circ: R \times G \to G$ is a binary operation.
Then the ring $\struct {R, +_R, \times_R}$ is called the scalar ring of $\struct {G, +_G, \circ}_R$.
If the scalar ring is understood, then $\struct {G, +_G, \circ}_R$ can be rendered $\struct {G, +_G, \circ}$.
Scalar
The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called scalars.
Zero Scalar
The zero of the scalar ring is called the zero scalar and usually denoted $0$, or, if it is necessary to distinguish it from the identity of $\struct {G, +_G}$, by $0_R$.
Scalar (or Base) Field
In linear algebra, it is usually referred to on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a base field:
Let $\struct {G, +_G, \circ}_K$ be a vector space, where:
- $\struct {K, +_K, \times_K}$ is a field
- $\struct {G, +_G}$ is an abelian group $\struct {G, +_G}$
- $\circ: K \times G \to G$ is a binary operation.
Then the field $\struct {K, +_K, \times_K}$ is called the base field of $\struct {G, +_G, \circ}_K$.