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 $$\left({G, +_G, \circ}\right)$$.