# Definition:Scalar Ring/Scalar Multiplication

## 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.

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

The operation $\circ: R \times S \to S$ is called scalar multiplication.

## Also known as

The source 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics refers to this as exterior multiplication.