Definition:Unitary Module

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.

$(UM \, 1)$	$:$	$\displaystyle \forall \lambda \in R: \forall x, y \in G:$	$\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 $

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.

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