Definition:Scalar Ring/Scalar

Definition

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 elements of the scalar ring $\left({R, +_R, \times_R}\right)$ are called scalars.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 7.32$