Definition:R-Algebraic Structure

Definition
Let $$\left({R, +_R, \times_R}\right)$$ be a ring.

Let $$\left({S, \ast_1, \ast_2, \ldots, \ast_n}\right)$$ be an algebraic structure with $$n$$ operations.

Let $$\circ: R \times S \to S$$ be a binary operation.

Then $$\left({S, \ast_1, \ast_2, \ldots, \ast_n: \circ}\right)_R$$ is an $$R$$-algebraic structure with $$n$$ operations.

Or, if the number of operations in $$S$$ is either understood or general, simply an $$R$$-algebraic structure, and the structure can be denoted $$\left({S: \circ}\right)_R$$.

Scalar Multiplication
The operation $$\circ: R \times S \to S$$ is called scalar multiplication.