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, it is just called an $R$-algebraic structure, and the structure can be denoted $\left({S, \circ}\right)_R$.

Terminology
The operation $\circ$ is also known as an $R$-action by some authors.

Also see

 * Definition:Scalar Ring
 * Definition:Module
 * Definition:Vector Space


 * Definition:Algebra over Ring
 * Definition:Algebra over Field