# Definition:Left Module Axioms

## Definition

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

A left module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ which satisfies the following conditions:

 $(M \, 1)$ $:$ Scalar Multiplication (Left) Distributes over Module Addition $\displaystyle \forall \lambda \in R: \forall x, y \in G:$ $\displaystyle \lambda \circ \paren {x +_G y}$ $\displaystyle =$ $\displaystyle \paren {\lambda \circ x} +_G \paren {\lambda \circ y}$ $(M \, 2)$ $:$ Scalar Multiplication (Right) Distributes over Scalar Addition $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle \paren {\lambda +_R \mu} \circ x$ $\displaystyle =$ $\displaystyle \paren {\lambda \circ x} +_G \paren {\mu \circ x}$ $(M \, 3)$ $:$ Associativity of Scalar Multiplication $\displaystyle \forall \lambda, \mu \in R: \forall x \in G:$ $\displaystyle \paren {\lambda \times_R \mu} \circ x$ $\displaystyle =$ $\displaystyle \lambda \circ \paren {\mu \circ x}$

These stipulations are called the left module axioms.

## Also known as

Some sources do not distinguish between a left module and a right module, and instead refer to these axioms as the module axioms.