Definition:Scalar Ring/Scalar Multiplication

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Let $\left({S, *_1, *_2, \ldots, *_n, \circ}\right)_R$ be an $R$-algebraic structure with $n$ operations, where:

$\left({R, +_R, \times_R}\right)$ is a ring
$\left({S, *_1, *_2, \ldots, *_n}\right)$ is an algebraic structure with $n$ operations
$\circ: R \times S \to S$ is a binary operation.

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

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.