Definition:Scalar Multiplication/R-Algebraic Structure

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
Some sources refer to this as exterior multiplication.