# Definition:Unitary Module

## 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:

 $(\text {UM} 1)$ $:$ $\displaystyle \forall \lambda \in R: \forall x, y \in G:$ $\displaystyle \lambda \circ \paren {x +_G y} = \paren {\lambda \circ x} +_G \paren {\lambda \circ y}$ $(\text {UM} 2)$ $:$ $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle \paren {\lambda +_R \mu} \circ x = \paren {\lambda \circ x} +_G \paren {\mu \circ x}$ $(\text {UM} 3)$ $:$ $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle \paren {\lambda \times_R \mu} \circ x = \lambda \circ \paren {\mu \circ x}$ $(\text {UM} 4)$ $:$ $\displaystyle \forall x \in G:$ $\displaystyle 1_R \circ x = x$

## Also known as

A unitary module is also known as a unital module. A unitary module over $R$ can also be referred to as a unitary $R$-module.

## Also see

• Results about unitary modules can be found here.