%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: R-module
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%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\subsection{R-Module and left/right module definitions}
\begin{definition}
Consider a ring $R$ with \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html}. Then a \emph{left module} $M_L$ over $R$ is defined as a set with two binary \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html},
$$+: M_L \times M_L \longrightarrow M_L$$ and $$\bullet : R \times M_L \longrightarrow M_L,$$ such that
\begin{enumerate}
\item $(\u+\v)+\w = \u+(\v+\w)$ for all $\u,\v,\w \in M_L$
\item $\u+\v=\v+\u$ for all $\u,\v\in M_L$
\item There exists an element $\0 \in M_L$ such that $\u+\0=\u$ for all $\u \in M_L$
\item For any $\u \in M_L$, there exists an element $\v \in M_L$ such that $\u+\v=\0$
\item $a \bullet (b \bullet \u) = (a \bullet b) \bullet \u$ for all $a,b \in R$ and $\u \in M_L$
\item $a \bullet (\u+\v) = (a \bullet\u) + (a \bullet \v)$ for all $a \in R$ and $\u,\v \in M_L$
\item $(a + b) \bullet \u = (a \bullet \u) + (b \bullet \u)$ for all $a,b \in R$ and $\u \in M_L$
\end{enumerate}
A right module $M_R$ is analogously defined to $M_L$ except for two things that are different in its definition:
\begin{enumerate}
\item the \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} ``$\bullet$'' goes from $M_R \times R$ to $M_R,$ and
\item the \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html} multiplication operations act on the right of the elements.
\end{enumerate}
\end{definition}
\begin{definition}
An \emph{R-module} generalizes the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of module to $n$-objects by employing Mitchell's definition of a ``ring with n-objects'' $R_n$; thus an \emph{$R$-module} is in fact an $R_n$ module with this notation.
\end{definition}
\subsection{Remarks}
One can define the \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} of left- and - right R-modules, whose \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} 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).
\end{document}
