\begin{definition}[Definition:Scalar Ring/Zero Scalar]
Let $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ be an $R$-algebraic structure with $n$ operations, where:
:$\struct {R, +_R, \times_R}$ is a ring
:$\struct {S, *_1, *_2, \ldots, *_n}$ is an algebraic structure with $n$ operations
:$\circ: R \times S \to S$ is a binary operation.
Let $\struct {R, +_R, \times_R}$ be the scalar ring of $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$.
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 $\struct {G, +_G}$, by $0_R$.
\end{definition}