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.

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

Also see

 * Scalar Ring
 * Module
 * Vector Space