Definition:Module/Left and Right Modules

Note
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 actions are commutative.

Also see

 * Definition:Left Module
 * Definition:Right Module
 * Definition:Module
 * Definition:Scalar Ring of Module
 * Basic Results about Modules