# Definition:Scalar Ring

## Definition

Let $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ be an $R$-algebraic structure with $n$ operations, where:

- $\struct {R, +_R, \times_R}$ is a ring

- $\struct {S, *_1, *_2, \ldots, *_n}$ is an algebraic structure with $n$ operations

- $\circ: R \times S \to S$ is a binary operation.

Then the ring $\struct {R, +_R, \times_R}$ is called the scalar ring of $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$.

If the scalar ring is understood, then $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ can be rendered $\struct {S, *_1, *_2, \ldots, *_n, \circ}$.

### 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 $0_R$.

## Definition for Module

Let $\struct {G, +_G, \circ}_R$ be a module, where:

- $\struct {R, +_R, \times_R}$ is a ring

- $\struct {G, +_G}$ 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 $\struct {G, +_G, \circ}_R$.

## Definition for Unitary Module

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 $\struct {G, +_G, \circ}_R$.

## Definition for Vector Space over Division Ring

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**.

## Scalar 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 **scalar field** of $\struct {G, +_G, \circ}_K$.