Definition:Unitary Module

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Definition

Let $\left({R, +_R, \times_R}\right)$ be a ring with unity whose unity is $1_R$.

Let $\left({G, +_G}\right)$ be an abelian group.


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

\((UM \, 1)\)   $:$     \(\displaystyle \forall \lambda \in R: \forall x, y \in G:\) \(\displaystyle \lambda \circ \paren {x +_G y} = \paren {\lambda \circ x} +_G \paren {\lambda \circ y} \)             
\((UM \, 2)\)   $:$     \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) \(\displaystyle \paren {\lambda +_R \mu} \circ x = \paren {\lambda \circ x} +_G \paren {\mu \circ x} \)             
\((UM \, 3)\)   $:$     \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) \(\displaystyle \paren {\lambda \times_R \mu} \circ x = \lambda \circ \paren {\mu \circ x} \)             
\((UM \, 4)\)   $:$     \(\displaystyle \forall x \in G:\) \(\displaystyle 1_R \circ x = x \)             


Also known as

A unitary module is also known as a unital module. A unitary module over $R$ can also be referred to as a unitary $R$-module.


Also see

  • Results about unitary modules can be found here.


Sources