# Definition:R-Algebraic Structure

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## Definition

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n}$ be an algebraic structure with $n$ operations.

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

Then $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ 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 $\struct {S, \circ}_R$.

## Also known as

An **$R$-algebraic structure** is also known as an **algebraic structure over (the ring) $R$**, but this terminology is not as clear, and so is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.

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

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules