Definition:Scalar Ring/Zero Scalar

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.

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

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 $0_R$.