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\begin{document}

\noindent
{\large \bf Additional notes \#2.}

\bigskip

Read Sections 10.1--3.

\bigskip

\noindent
{\bf Examples of Modules.}

\bigskip

\begin{enumerate}
\item[(1)] If ${\BK}$ is a division ring, then
${\BK}$-modules coincide with ${\BK}$-vector spaces.
\bigskip
\item[(2)] If $\m V$ is an $n$-dimensional 
$\BK$-vector space, then $\m V$ is 
an $M_n({\BK})$-module.
\bigskip
\item[(3)] If $\m a$ is an abelian
group, then $\m a$ is a $\BZ$-module:
let $n(a) = \underbrace{a + \cdots + a}_{\mbox{$n$ $a$'s}}$.
%\bigskip
\item[(4)] If $\m a$ is an abelian group, then 
$\m a$ is an ${\rm End}(\m a)$-module.
\bigskip
\item[(5)] If $\m m$ is an $\m s$-module and 
$\varphi:\m r\rightarrow \m s$ is a ring homomorphism, 
then $\m m$ is an $\m r$-module: let $r(m) = \varphi(r)(m)$.
\bigskip
\item[(6)] Let $\m g$ be a group. A {\it $\m g$-module}
is an algebra $\m m = \lb M; +, -, 0, \{g(x)\;|\;g\in G\}\rb$
for which 
\begin{enumerate}
\item[(a)] $g(m+n) = g(m)+g(n)$, 
\item[(b)] $g(h(m)) = (gh)(m)$, and
\item[(c)] $1(m) = m$.
\end{enumerate}
Equivalently, a $\m g$-module (in this definition)
is a ${\BZ}[\m g]$-module (in the previous definition).
\item[(7)] Assume that $\m n\triangleleft \m g$ and
that $\m n$ is abelian. Then $\m n$ is a $\m g$-module:
$g(n) = gng^{-1}$.
\item[(8)] If $I$ is an ideal of a ring $\m r$, then it is
both a left and a right $\m r$-module (it is an
{\it $(\m r,\m r)$-bimodule}): $r(i) = ri$, $(i)r = ir$.
\item[(9)] Let $\m v$ be an $\BF$-vector space
and let $T$ be a linear transformation of $\m v$.
Then $\m v$ is an ${\BF}[x]$-module:
$p(x)(v) = p(T)(v)$.
\end{enumerate}

\bigskip\bigskip\bigskip

\noindent
Exercises.
\begin{enumerate}
\item[(1)] Let $\m m$ be an $\m r$-module.
Show that the function $\varphi: \m r\rightarrow
{\rm End}(\lb M; +, -, 0\rb): r\mapsto r(x)$
is a ring homomorphism.
\item[(2)] Let $\m v$ be the Klein group. Then
$\m v$ is a normal abelian subgroup of each of 
$\m v < \m a_4 < \m s_4$. Show that 
\begin{enumerate}
\item[(a)] The $\m v$-module structure on $\m v$ is
that of a 2-dimensional vector space over the 2-element field,
${\BF}_2$.
\item[(b)] The $\m a_4$-module structure on $\m v$ is
that of a 1-dimensional vector space over the 4-element field,
${\BF}_4$.
\item[(c)] The $\m s_4$-module structure on $\m v$ is
that of $M_2({\BF}_2)$ acting on a 
2-dimensional vector space over ${\BF}_2$
\end{enumerate}
\end{enumerate}
\end{document}

