# Category:Unitary Modules

This category contains results about Unitary Modules.
Definitions specific to this category can be found in Definitions/Unitary Modules.

== Definition == <onlyinclude> 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 is either a unitary left module or a unitary right module, the type is unspecified:

### Unitary Left Module

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

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

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

 $(\text {UM} 1)$ $:$ Scalar Multiplication (Left) Distributes over Module Addition $\ds \forall \lambda \in R: \forall x, y \in G:$ $\ds \lambda \circ \paren {x +_G y}$ $\ds =$ $\ds \paren {\lambda \circ x} +_G \paren {\lambda \circ y}$ $(\text {UM} 2)$ $:$ Scalar Multiplication (Right) Distributes over Scalar Addition $\ds \forall \lambda, \mu \in R: \forall x \in G:$ $\ds \paren {\lambda +_R \mu} \circ x$ $\ds =$ $\ds \paren {\lambda \circ x} +_G \paren {\mu \circ x}$ $(\text {UM} 3)$ $:$ Associativity of Scalar Multiplication $\ds \forall \lambda, \mu \in R: \forall x \in G:$ $\ds \paren {\lambda \times_R \mu} \circ x$ $\ds =$ $\ds \lambda \circ \paren {\mu \circ x}$ $(\text {UM} 4)$ $:$ Unity of Scalar Ring $\ds \forall x \in G:$ $\ds 1_R \circ x$ $\ds =$ $\ds x$

### Unitary Right Module

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

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

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

 $(\text {URM} 1)$ $:$ Scalar Multiplication Right Distributes over Module Addition $\ds \forall \lambda \in R: \forall x, y \in G:$ $\ds \paren {x +_G y} \circ \lambda$ $\ds =$ $\ds \paren {x \circ \lambda} +_G \paren {y \circ \lambda}$ $(\text {URM} 2)$ $:$ Scalar Multiplication Left Distributes over Scalar Addition $\ds \forall \lambda, \mu \in R: \forall x \in G:$ $\ds x \circ \paren {\lambda +_R \mu}$ $\ds =$ $\ds \paren {x \circ \lambda} +_G \paren {x\circ \mu}$ $(\text {URM} 3)$ $:$ Associativity of Scalar Multiplication $\ds \forall \lambda, \mu \in R: \forall x \in G:$ $\ds x \circ \paren {\lambda \times_R \mu}$ $\ds =$ $\ds \paren {x \circ \lambda} \circ \mu$ $(\text {URM} 4)$ $:$ Unity of Scalar Ring $\ds \forall x \in G:$ $\ds x \circ 1_R$ $\ds =$ $\ds x$

## 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.

## Subcategories

This category has the following 4 subcategories, out of 4 total.

## Pages in category "Unitary Modules"

The following 13 pages are in this category, out of 13 total.