PlanetPhysics/R Module

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R-Module and left/right module definitions

Consider a ring $R$ with [[../Cod/|identity]]. Then a left module $M_L$ over $R$ is defined as a set with two binary [[../Cod/|operations]], $+:M_L\times M_L⟶M_L$ and $\bullet :R\times M_L⟶M_L,$ such that

1. Failed to parse (unknown function "\u"): {\displaystyle (\u+\v)+\w = \u+(\v+\w)} for all Failed to parse (unknown function "\u"): {\displaystyle \u,\v,\w \in M_L}
2. Failed to parse (unknown function "\u"): {\displaystyle \u+\v=\v+\u} for all Failed to parse (unknown function "\u"): {\displaystyle \u,\v\in M_L}
3. There exists an element Failed to parse (syntax error): {\displaystyle \0 \in M_L} such that Failed to parse (unknown function "\u"): {\displaystyle \u+\0=\u} for all Failed to parse (unknown function "\u"): {\displaystyle \u \in M_L}
4. For any Failed to parse (unknown function "\u"): {\displaystyle \u \in M_L} , there exists an element Failed to parse (unknown function "\v"): {\displaystyle \v \in M_L} such that Failed to parse (unknown function "\u"): {\displaystyle \u+\v=\0}
5. Failed to parse (unknown function "\u"): {\displaystyle a \bullet (b \bullet \u) = (a \bullet b) \bullet \u} for all $a,b\in R$ and Failed to parse (unknown function "\u"): {\displaystyle \u \in M_L}
6. Failed to parse (unknown function "\u"): {\displaystyle a \bullet (\u+\v) = (a \bullet\u) + (a \bullet \v)} for all $a\in R$ and Failed to parse (unknown function "\u"): {\displaystyle \u,\v \in M_L}
7. Failed to parse (unknown function "\u"): {\displaystyle (a + b) \bullet \u = (a \bullet \u) + (b \bullet \u)} for all $a,b\in R$ and Failed to parse (unknown function "\u"): {\displaystyle \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 [[../TrivialGroupoid/|morphism]] "$\bullet$" goes from $M_R\times R$ to $M_R,$ and

1. the [[../Vectors/|scalar]] multiplication operations act on the right of the elements.

An R-module generalizes the [[../PreciseIdea/|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 [[../Cod/|categories]] of left- and - right R-modules, whose [[../TrivialGroupoid/|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).