# Definition:Scalar

## Definition

### R-Algebraic Structure

Let $\struct {R, +_R, \times_R}$ be the scalar ring of an $R$-algebraic structure $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$.

The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called **scalars**.

### Module

Let $\struct {R, +_R, \times_R}$ be the scalar ring of a module $\struct {G, +_G, \circ}_R$.

The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called **scalars**.

### Vector Space

Let $\struct {K, +_K, \times_K}$ be the scalar field of a vector space $\struct {G, +_G, \circ}_K$.

The elements of the scalar field $\struct {K, +_K, \times_K}$ are called **scalars**.

### Scalar (Matrix Theory)

Let $\map \MM {m, n}$ be a matrix space of order $m \times n$ over an underlying structure $R$.

The elements of $R$ are referred to as **scalars**.

### Scalar Quantity

A **scalar quantity** is a real-world concept that needs for its model a mathematical object which contains only one (usually numeric) component.

## Sources

- 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $1$. Scalar and Vector Quantities