# PlanetPhysics/R Module

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### R-Module and left/right module definitionsEdit

Consider a ring with identity. Then a *left module* over is defined as a set with two binary operations,
and such that

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A right module is analogously defined to except for two things that are different in its definition:

- the morphism " " goes from to and

- the scalar multiplication operations act on the right of the elements.

An *R-module* generalizes the concept of module to -objects by employing Mitchell's definition of a "ring with n-objects" ; thus an * -module* is in fact an module with this notation.

### RemarksEdit

One can define the categories of left- and - right R-modules, whose objects are, respectively, left- and - right R-modules, and whose arrows are R-module morphisms.

If the ring is commutative one can prove that the category of left --modules and the category of right --modules are equivalent (in the sense of an equivalence of categories, or categorical equivalence).