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 $\left({S, *_1, *_2, \ldots, *_n, \circ}\right)$.

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.