Category:Unitary Modules

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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.


Sources