# Definition:Scalar Multiplication/R-Algebraic Structure

< Definition:Scalar Multiplication(Redirected from Definition:Scalar Ring/Scalar Multiplication)

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## 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

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

## Also known as

Some sources refer to **scalar multiplication** as **exterior multiplication**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules