PlanetPhysics/R Module

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R-Module and left/right module definitions

Consider a ring ${\displaystyle R}$  with identity. Then a left module ${\displaystyle M_{L}}$  over ${\displaystyle R}$  is defined as a set with two binary operations, ${\displaystyle +:M_{L}\times M_{L}\longrightarrow M_{L}}$  and ${\displaystyle \bullet :R\times M_{L}\longrightarrow M_{L},}$  such that

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

A right module ${\displaystyle M_{R}}$  is analogously defined to ${\displaystyle M_{L}}$  except for two things that are different in its definition:

1. the morphism "${\displaystyle \bullet }$ " goes from ${\displaystyle M_{R}\times R}$  to ${\displaystyle M_{R},}$  and
1. the scalar multiplication operations act on the right of the elements.

An R-module generalizes the concept of module to ${\displaystyle n}$ -objects by employing Mitchell's definition of a "ring with n-objects" ${\displaystyle R_{n}}$ ; thus an ${\displaystyle R}$ -module is in fact an ${\displaystyle R_{n}}$  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 ${\displaystyle R}$  is commutative one can prove that the category of left ${\displaystyle R}$ --modules and the category of right ${\displaystyle R}$ --modules are equivalent (in the sense of an equivalence of categories, or categorical equivalence).