Definition:Scalar Ring/Zero Scalar

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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 $0_R$.