Definition:Right Module

Definition

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

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

A right module over $R$ is an $R$-algebraic structure $\struct {G, +_G, \circ}_R$ with one operation $\circ$, the (right) ring action, which satisfies the right module axioms:

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

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:Left Module
• Definition:Module

• Results about modules can be found here.

Sources

• 2003: P.M. Cohn: Basic Algebra: Groups, Rings and Fields ... (next): Chapter $4$: Rings and Modules: $\S 4.1$: The Definitions Recalled