Definition:Scalar Ring

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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


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 $\struct {R, +_R, \times_R}$ 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