# 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.