Axiom:Unitary Module Axioms

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 following conditions:

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

These stipulations are called the unitary module axioms.

Also see

• {{Module-axiom}}