# Definition:Scalar Ring/Unitary Module

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## 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 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 $\struct {G, +_G, \circ}_K$.