PlanetPhysics/R Module

\newcommand{\sqdiagram}[9]{Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

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. 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 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).