PlanetPhysics/R Module

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R-Module and left/right module definitions
Consider a ring $$R$$ with identity. Then a left module $$M_L$$ over $$R$$ is defined as a set with two binary operations, $$+: M_L \times M_L \longrightarrow M_L$$ and $$\bullet : R \times M_L \longrightarrow M_L,$$ such that


 * 1) $$(\u+\v)+\w = \u+(\v+\w)$$ for all $$\u,\v,\w \in M_L$$
 * 2) $$\u+\v=\v+\u$$ for all $$\u,\v\in M_L$$
 * 3) There exists an element $$\0 \in M_L$$ such that $$\u+\0=\u$$ for all $$\u \in M_L$$
 * 4) For any $$\u \in M_L$$, there exists an element $$\v \in M_L$$ such that $$\u+\v=\0$$
 * 5) $$a \bullet (b \bullet \u) = (a \bullet b) \bullet \u$$ for all $$a,b \in R$$ and $$\u \in M_L$$
 * 6) $$a \bullet (\u+\v) = (a \bullet\u) + (a \bullet \v)$$ for all $$a \in R$$ and $$\u,\v \in M_L$$
 * 7) $$(a + b) \bullet \u = (a \bullet \u) + (b \bullet \u)$$ for all $$a,b \in R$$ and $$\u \in M_L$$

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


 * 1) the morphism "$$\bullet$$" goes from $$M_R \times R$$ to $$M_R,$$ and


 * 1) the scalar multiplication operations act on the right of the elements.

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

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