Category:Definitions/Unitary Modules

This category contains definitions related to Unitary Modules.
Related results can be found in Category: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:

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

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

$(\text {UM} 1)$	$:$	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 $

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

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

$(\text {URM} 1)$	$:$	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 $

A unitary module is also known as a unital module.

This category has the following 5 subcategories, out of 5 total.

The following 7 pages are in this category, out of 7 total.

\((\text {UM} 1)\)	$:$	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	\(\ds \forall x \in G:\)	\(\ds 1_R \circ x \)	\(\ds = \)	\(\ds x \)

\((\text {URM} 1)\)	$:$	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	\(\ds \forall x \in G:\)	\(\ds x \circ 1_R \)	\(\ds = \)	\(\ds x \)