RModule 2009-01-31 19:29:04 2009-01-31 19:51:56 Definition category of left R-modules categories of right R-modules left- and - right R-modules categories of left- and - right R-modules % this is the default PlanetPhysics preamble. \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate,color} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} \renewcommand{\u}{\mathbf{u}} \renewcommand{\v}{\mathbf{v}} \newcommand{\w}{\mathbf{w}} \newcommand{\0}{\mathbf{0}} \def\blue{\textcolor{blue}} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{\mathcal G} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\midsqn}[1]{\ar@{}[dr]|{#1}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \subsection{R-Module and module definitions} \begin{definition} Consider a ring $R$ with identity. Then a \emph{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 \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 morphism ``$\bullet$'' goes from $M_R \times R$ to $M_R,$ and \item the scalar multiplication operations act on the right of the elements. \end{enumerate} \end{definition} \textbf{Remarks} One can define the categories of left- and - right R-modules, whose objects are, respectively, left- and - right R-modules, and 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). \begin{definition} An \emph{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 \emph{$R$-module} is in fact an $R_n$ module with this notation. \end{definition}