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\begin{document}

\begin{center}
{\bf\large  ALGEBRA TEST \#1}
\end{center}

\vspace{1in}

This exam is due Wednesday, February 20.
Do {\bf four} of the problems.
You may use your book, but you may not
communicate with others concerning the exam.
In order to receive full credit your answer must be 
{\bf complete}, {\bf legible} and {\bf correct}. 

\vfill

\begin{center}
I have neither given nor received aid on this exam.
\end{center}

\vspace{.5in}

\begin{flushright}
{\bf Name:\underline{\hspace{2.5in}}}
\end{flushright}

\break

1. Say that a function $f:X\rightarrow Y$ between metric spaces
is {\it nonexpansive} if $d_Y(f(a),f(b))\leq d_X(a,b)$,
where $d_X$ and $d_Y$ are the distance
functions of $X$ and $Y$. Show that the category whose objects
are metric spaces and whose morphisms are all nonexpansive
functions has
%is a category with 
finite products but not infinite products.

\vfill

2. Show that the category of compact topological spaces
(morphisms = all continuous functions) does not have
free objects over any infinite set.

\vfill

3. Which abelian group is presented by
$\lb x, y\;|\; 3x+5y=0, 7x+10y=0\rb?$


\vfill

4. How large is the ring presented by
$\lb x\;|\; 3=0, x^2+1=0\rb?$
Is it a field?

\vfill

5. Let $\{\m m_i\;|\;i\in I\}$ be an
infinite family of nonzero
$\m r$-modules. Show that
\[|\bigoplus_{i\in I} \m m_i| = |I| \cdot \sup\{|M_i|\;|\;i\in I\}.\]
%\[|I|, \sup\{|M_i|\;|\;i\in I\}\leq |\bigoplus_{i\in I} \m m_i|
%\leq |I| \cdot \sup\{|M_i|\;|\;i\in I\}.\]

\vfill

6. Let $\m m$ be an $\m r$-module. Show that 
$I = \{r\in R\;|\;rm=0\;\mbox{\rm for all $m\in M$}\}$
is an ideal of $\m r$ (called the {\it annihilator}
of $\m m$). Show that there is 
a natural way to view $\m m$ as an 
$\m r/I$-module.

\vfill

7. Let $\m s$ be a simple $\m r$-module. Show that 
$\m d = {\rm End}_{\m r}(\m s)$ is a division ring with 
respect to the operations of composition, pointwise
addition and negation, the zero function and the identity
function.

\vfill

8. Let $\m r = M_n({\BF})$ for some field ${\BF}$. Show that
the 1-generated free module ${}_{\m r}{\m r}$ is 
isomorphic to a direct sum of $n$ isomorphic simple modules.


\vfill

\end{document}
