Mathematical Structures: Modules over a ring

# Modules over a ring

Difference (from prior major revision) (no other diffs)

Changed: 1,120c1,120
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Modules over a ring} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Modules_over_a_ring}{edit} \abbreviation{RMod} \begin{definition} A \emph{module over a \href{Rings_with_identity.pdf}{rings with identity}} $\mathbf{R}$ is a structure $\mathbf{A}=\langle A,+,-,0,f_r\ (r\in R)\rangle$ such that $\langle A,+,-,0\rangle$ is an \href{Abelian_groups.pdf}{abelian groups} $f_r$ preserves addition: $f_r(x+y)=f_r(x)+f_r(y)$ $f_{1}$ is the identity map: $f_{1}(x)=x$ $f_{r+s}(x))=f_r(x)+f_s(x)$ $f_{r\circ s}(x)=f_r(f_s(x))$ Remark: $f_r$ is called \emph{scalar multiplication by $r$}, and $f_r(x)$ is usually written simply as $rx$. \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be modules over a ring $\mathbf{R}$. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a group homomorphism and preserves all $f_r$: $h(f_r(x))=f_r(h(x))$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & variety\\\hline Equational theory & \\\hline Quasiequational theory & \\\hline First-order theory & \\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline Congruence distributive & no\\\hline Congruence modular & yes\\\hline Congruence n-permutable & yes, $n=2$\\\hline Congruence regular & yes\\\hline Congruence uniform & yes\\\hline Congruence extension property & yes\\\hline Definable principal congruences & no\\\hline Equationally def. pr. cong. & no\\\hline Amalgamation property & \\\hline Strong amalgamation property & \\\hline Epimorphisms are surjective & \\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \end{subclasses} \begin{superclasses}\ \href{Abelian_groups.pdf}{Abelian groups} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Modules over a ring} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Modules_over_a_ring}{edit} \abbreviation{RMod} \begin{definition} A \emph{module over a \href{Rings_with_identity.pdf}{rings with identity}} $\mathbf{R}$ is a structure $\mathbf{A}=\langle A,+,-,0,f_r\ (r\in R)\rangle$ such that $\langle A,+,-,0\rangle$ is an \href{Abelian_groups.pdf}{abelian groups} $f_r$ preserves addition: $f_r(x+y)=f_r(x)+f_r(y)$ $f_{1}$ is the identity map: $f_{1}(x)=x$ $f_{r+s}(x))=f_r(x)+f_s(x)$ $f_{r\circ s}(x)=f_r(f_s(x))$ Remark: $f_r$ is called \emph{scalar multiplication by $r$}, and $f_r(x)$ is usually written simply as $rx$. \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be modules over a ring $\mathbf{R}$. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a group homomorphism and preserves all $f_r$: $h(f_r(x))=f_r(h(x))$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & variety\\\hline Equational theory & \\\hline Quasiequational theory & \\\hline First-order theory & \\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline Congruence distributive & no\\\hline Congruence modular & yes\\\hline Congruence n-permutable & yes, $n=2$\\\hline Congruence regular & yes\\\hline Congruence uniform & yes\\\hline Congruence extension property & yes\\\hline Definable principal congruences & no\\\hline Equationally def. pr. cong. & no\\\hline Amalgamation property & \\\hline Strong amalgamation property & \\\hline Epimorphisms are surjective & \\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \end{subclasses} \begin{superclasses}\ \href{Abelian_groups.pdf}{Abelian groups} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %

http://mathcs.chapman.edu/structuresold/files/Modules_over_a_ring.pdf
%%run pdflatex

%


\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Modules over a ring}

\abbreviation{RMod}
\begin{definition}
A \emph{module over a \href{Rings_with_identity.pdf}{rings with identity}} $\mathbf{R}$ is a structure $\mathbf{A}=\langle A,+,-,0,f_r\ (r\in R)\rangle$ such that

$\langle A,+,-,0\rangle$ is an \href{Abelian_groups.pdf}{abelian groups}

$f_r$ preserves addition:
$f_r(x+y)=f_r(x)+f_r(y)$

$f_{1}$ is the identity map:  $f_{1}(x)=x$

$f_{r+s}(x))=f_r(x)+f_s(x)$

$f_{r\circ s}(x)=f_r(f_s(x))$

Remark:
$f_r$ is called \emph{scalar multiplication by $r$}, and $f_r(x)$ is usually written simply as $rx$.

\end{definition}
\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be modules over a ring $\mathbf{R}$.
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a group homomorphism and preserves all $f_r$:

$h(f_r(x))=f_r(h(x))$
\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & no\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & no\\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\end{subclasses}
\begin{superclasses}\

\href{Abelian_groups.pdf}{Abelian groups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%