Definition:Unitary Module

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In particular: Distinguish between a unitary left module and unitary right module
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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:

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


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

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