Mathematical Structures: Modules over a ring

# Modules over a ring

http://mathcs.chapman.edu/structuresold/files/Modules_over_a_ring.pdf
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\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/}
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\begin{subclasses}\

\end{subclasses}
\begin{superclasses}\

\href{Abelian_groups.pdf}{Abelian groups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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