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:

• Left Module over Commutative Ring induces Right Module
• Right Module over Commutative Ring induces Left Module.

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

• Left Module Does Not Necessarily Induce Right Module over Ring
• Right Module Does Not Necessarily Induce Left Module over Ring

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

From:

• Left Module over Ring Induces Right Module over Opposite Ring
• Right Module over Ring Induces Left Module over Opposite Ring

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

From:

• Left Module induces Right Module over same Ring iff Actions are Commutative
• Right Module induces Left Module over same Ring iff Actions are Commutative

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

Also see

• Definition:Right Module
• Definition:Module

• Results about modules can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): module
• 2003: P.M. Cohn: Basic Algebra: Groups, Rings and Fields ... (previous) ... (next): Chapter $4$: Rings and Modules: $\S 4.1$: The Definitions Recalled
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): module