Definition:Scalar Ring

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.

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

If the scalar ring is understood, then $\left({S, *_1, *_2, \ldots, *_n, \circ}\right)_R$ can be rendered $\left({S, *_1, *_2, \ldots, *_n, \circ}\right)$.

Scalar

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

Scalar Multiplication

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

Zero Scalar

The zero of the scalar ring is called the zero scalar and usually denoted $0$, or, if it is necessary to distinguish it from the identity of $\left({G, +_G}\right)$, by $0_R$.

Definition for Module

The same definition applies when $\left({S, *_1, *_2, \ldots, *_n}\right)$ is an abelian group $\left({G, +_G}\right)$.

In this case, $\left({G, +_G, \circ}\right)_R$ is a module.

The same definition also applies when $\left({G, +_G, \circ}\right)_R$ is a unitary module, but in this latter case note that $\left({R, +_R, \times_R}\right)$ is a ring with unity.

Scalar Field

Let $\left({G, +_G, \circ}\right)_K$ be a vector space, where:

• $\left({K, +_K, \times_K}\right)$ is a field

• $\left({G, +_G}\right)$ is an abelian group $\left({G, +_G}\right)$

• $\circ: K \times G \to G$ is a binary operation.

Then the field $\left({K, +_K, \times_K}\right)$ is called the scalar field of $\left({G, +_G, \circ}\right)_K$.

If the scalar field is understood, then $\left({G, +_G, \circ}\right)_K$ can be rendered $\left({G, +_G, \circ}\right)$.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$