Definition:Unitary Module Axioms

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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 following conditions:

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

These stipulations are called the unitary module axioms.

Also see