Definition:R-Algebraic Structure

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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

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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

• Results about $R$-algebraic structures can be found here.

Sources

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