Category:Unitary Modules

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

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

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} = \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 $

Subcategories

This category has the following 2 subcategories, out of 2 total.

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

\((\text {UM} 1)\)	$:$	Scalar Multiplication (Left) Distributes over Module Addition	\(\ds \forall \lambda \in R: \forall x, y \in G:\)	\(\ds \lambda \circ \paren {x +_G y} = \paren {\lambda \circ x} +_G \paren {\lambda \circ y} \)
\((\text {UM} 2)\)	$:$	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	\(\ds \forall x \in G:\)	\(\ds 1_R \circ x = x \)