Definition:Module/Left and Right Modules

Left and 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 actions are commutative.

Given the remarks above, a theorem about modules, where the type of module is unspecified, is in effect two theorems:
 * one about left modules where all modules are left modules
 * the other about right modules where all modules are right modules.

The theorem may not be true for a mix of left and right modules, unless the modules are over a commutative ring.

The proof of such a theorem is generally given for one type of module only.

The proof for the other type is proved similarly with the scalar applied on the other side.

For example, see Direct Product of Modules is Module.

Where a theorem does involve a mix of left and right modules it is necessary to explicitly identify which modules are the left modules and which are the right modules.

For example, see Left Module over Ring Induces Right Module over Opposite Ring.

Also see

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