# Definition:Left Module

## 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 $\struct {G, +_G, \circ}_R$ with one operation $\circ$, the (left) ring action, which satisfies the left module axioms:

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

## Left vs Right Modules

In the case of a commutative ring, the difference between a left module and a right module is little more than a notational difference. See:

But this is not the case for a ring that is not commutative. From:

it is known that it is not sufficient to simply reverse the scalar multiplication to get a module of the other ‘side’.

From:

to obtain a module of the other ‘side’ it is, in general, also necessary to reverse the product of the ring.

From:

a left module induces a right module and vice-versa if and only if actions are commutative.

## Also see

• Results about modules can be found here.