Definition:Scalar Ring

Definition
Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ be an $R$-algebraic structure with $n$ operations, where:


 * $\left({R, +_R, \times_R}\right)$ is a ring


 * $\left({S, \ast_1, \ast_2, \ldots, \ast_n}\right)$ 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 $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$.

If the scalar ring is understood, then $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ can be rendered $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)$.

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 $\left({S, \ast_1, \ast_2, \ldots, \ast_n}\right)$ 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 an 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 $\left({G, +_G, \circ}\right)$.