# Definition:Unitary Module

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## Definition

Let $\struct {R, +_R, \times_R}$ be a ring with unity whose unity is $1_R$.

Let $\struct {G, +_G}$ be an abelian group.

A unitary module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ which satisfies the unitary module axioms:

\((\text {UM} 1)\) | $:$ | Scalar Multiplication (Left) Distributes over Module Addition | \(\ds \forall \lambda \in R: \forall x, y \in G:\) | \(\ds \lambda \circ \paren {x +_G y} = \paren {\lambda \circ x} +_G \paren {\lambda \circ y} \) | |||||

\((\text {UM} 2)\) | $:$ | Scalar Multiplication (Right) Distributes over Scalar Addition | \(\ds \forall \lambda, \mu \in R: \forall x \in G:\) | \(\ds \paren {\lambda +_R \mu} \circ x = \paren {\lambda \circ x} +_G \paren {\mu \circ x} \) | |||||

\((\text {UM} 3)\) | $:$ | Associativity of Scalar Multiplication | \(\ds \forall \lambda, \mu \in R: \forall x \in G:\) | \(\ds \paren {\lambda \times_R \mu} \circ x = \lambda \circ \paren {\mu \circ x} \) | |||||

\((\text {UM} 4)\) | $:$ | Unity of Scalar Ring | \(\ds \forall x \in G:\) | \(\ds 1_R \circ x = x \) |

## Also known as

A **unitary module over $R$** can also be referred to as a ** unitary $R$-module**.

A **unitary module** is also known as a **unital module**.

## Also see

- Results about
**unitary modules**can be found**here**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Module**