Definition:Left Module

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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’.


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


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.