\documentclass[12pt]{amsart}
\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb}
\pagestyle{headings}
\setlength{\textwidth}{36true pc}
\setlength{\headheight}{8true pt} % needed for 12pt documents
\setlength{\oddsidemargin}{15 truept}
\setlength{\evensidemargin}{15 truept}
\setlength{\textheight}{572true pt}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lm}[thm]{Lemma}
\newtheorem{prp}[thm]{Proposition}
\newtheorem{clm}[thm]{Claim}
%
\theoremstyle{definition}
\newtheorem{df}[thm]{Definition}
\newtheorem{exmp}[thm]{Example}
\newtheorem{prb}[thm]{Problem}
%
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
%
\numberwithin{equation}{section}
%
\newcommand{\thref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lmref}[1]{Lemma~\ref{#1}}
%
\DeclareMathOperator{\satf}{\models}
\DeclareMathOperator{\miff}{\Longleftrightarrow} 
                                           % math `if and only if'
\DeclareMathOperator{\mimplies}{\Longrightarrow} 
                                           % math `implies'
\DeclareMathOperator{\mimplied}{\Longleftarrow}  
                                           % math `implied by'
\newcommand{\set}[2]{\{#1\,:\,\text{#2}\}} % { x : ... }
\DeclareMathOperator{\id}{\textsl{id}}     % The identity map
\DeclareMathOperator{\miffdef}{\stackrel{def}{\miff}}
                                           % defining iff
\DeclareMathOperator{\eqdef}{\stackrel{def}{=}}
                                           % defining equation

\DeclareMathOperator{\neareq}{\stackrel{\ae}{=}}
\DeclareMathOperator{\aeqv}{\stackrel{ae}{\equiv}}
\DeclareMathOperator{\aeq}{\stackrel{ae}{=}}
%
\DeclareMathOperator{\BZ}{\mathbb{Z}} % integers
\DeclareMathOperator{\BQ}{\mathbb{Q}} % rationals
\DeclareMathOperator{\BR}{\mathbb{R}} % reals
\DeclareMathOperator{\BC}{\mathbb{C}} % complex numbers
\DeclareMathOperator{\BD}{\mathbb{D}} % 
\DeclareMathOperator{\BE}{\mathbb{E}} % 
\DeclareMathOperator{\BF}{\mathbb{F}} % 
\DeclareMathOperator{\BH}{\mathbb{H}} % 
\DeclareMathOperator{\BK}{\mathbb{K}} % 
\DeclareMathOperator{\BL}{\mathbb{L}} % 
\DeclareMathOperator{\re}{Re}         % real part
\DeclareMathOperator{\im}{Im}         % imaginary part
\DeclareMathOperator{\Ker}{Ker}
%
\DeclareMathOperator{\meet}{\wedge}
\DeclareMathOperator{\join}{\vee}
\DeclareMathOperator{\union}{\cup}
\DeclareMathOperator{\inter}{\cap}  % intersection
\DeclareMathOperator{\iso}{\cong}   % isomorphic
\DeclareMathOperator{\piso}{\simeq} % polynomially isomorphic
\newcommand{\Star}{^ *}
\DeclareMathOperator{\compose}{{\scriptstyle \circ}} % f o g
%
\newcommand{\wec}[1]{{\mathbf{#1}}}  % notation for vectors in algebras
%\newcommand{\wec}[1]{{\bar{#1}}} % which one do you prefer?
\newcommand{\wrel}[1]{\;#1\;}     % a\wrel{\alpha}b = a \alpha b
%              % this provides a flexible way to change the spacing
%
\newcommand{\m}[1]{{\mathbf{\uppercase{#1}}}}
\newcommand{\vr}[1]{{\mathcal{\uppercase {#1}}}}
%
\DeclareMathOperator{\Var}{\mathbf{V}}
\DeclareMathOperator{\Ho}{\mathbf{H}}
\DeclareMathOperator{\Su}{\mathbf{S}}
\DeclareMathOperator{\Pd}{\mathbf{P}}
\DeclareMathOperator{\Iso}{\mathbf{I}}
\DeclareMathOperator{\Ps}{\mathbf{P}_s}
\newcommand{\fin}[1]{{\mathbf{#1}}_{\text{\upshape fin}}}
%
\DeclareMathOperator{\Pol}{Pol}
\DeclareMathOperator{\Clo}{Clo}
\DeclareMathOperator{\Con}{Con}
\DeclareMathOperator{\Conb}{\mathbf{Con}} % Con(A) as a lattice
\DeclareMathOperator{\cg}{Cg}
\DeclareMathOperator{\sg}{Sg}
\newcommand{\Cg}[1]{{\cg}^{\m{#1}}}
\newcommand{\Sg}[1]{{\sg}^{\m{#1}}}
%
\newcommand{\M}[3]{\text{\upshape M}_{\m{#1}} (#2, #3)}
\newcommand{\minl}[1]{\text{\upshape Min}_{\m{#1}}}
%
\DeclareMathOperator{\typ}{typ}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Sep}{Sep}
\DeclareMathOperator{\ssb}{ssb}
\DeclareMathOperator{\sst}{sst}
%
\DeclareMathOperator{\utyp}{\mathbf{1}}% unary      
\DeclareMathOperator{\atyp}{\mathbf{2}}% affine     
\DeclareMathOperator{\btyp}{\mathbf{3}}% boolean    
\DeclareMathOperator{\ltyp}{\mathbf{4}}% lattice    
\DeclareMathOperator{\styp}{\mathbf{5}}% semilattice
\DeclareMathOperator{\ityp}{\mathbf{i}}
\DeclareMathOperator{\jtyp}{\mathbf{j}}
%
\DeclareMathOperator{\solv}{\stackrel{s}{\sim}}
\DeclareMathOperator{\ssolv}{\stackrel{ss}{\sim}}
%
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}
%
\newcommand{\lb}{\langle} % <...>
\newcommand{\rb}{\rangle}
\newcommand{\lv}{{[\![}}  % [[...]]
\newcommand{\rv}{{]\!]}}
%
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
%
\newcommand{\vep}{\varepsilon}
\newcommand{\di}{\diamond}
%


\begin{document}

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

\vspace{1in}

This exam is due Friday, February 16.
You are expected to complete three problems,
one from each group.
Clearly mark which problems are to be graded.

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

\noindent
{\bf Group 1.}

\medskip

1. Let $\m m$ be an $(\m r,\m s)$-bimodule
and let $\m n$ be an $(\m r,\m t)$-bimodule.
Explain how to make the set 
${\rm Hom}_{\m r}(\m m,\m n)$
of left $\m r$-module
homomorphisms into an $(\m s,\m t)$-bimodule.

\vfill

2. 
Let ${\BD}$ be a noncommutative division ring.
Suppose that $\varphi: \oplus^n {\BD} \rightarrow 
\oplus^m {\BD}$ is a homomorphism of {\it left}
${\BD}$-vector spaces, and that 
$\psi: \oplus^n {\BD} \rightarrow 
\oplus^m {\BD}$ is a homomorphism of {\it right}
${\BD}$-vector spaces. Explain how to find matrix
representations for $\varphi$ and $\psi$ so that
these homomorphisms correspond to 
(left or right) matrix multiplication.

\vfill

\noindent
{\bf Group 2.}

\medskip


3. Let $\vr V$ be the variety of groups of exponent 
four.\footnote{That is, the variety 
of groups satisfying $x^4 = 1$.}
Write a presentation $\lb G\;|\;R\rb$ for
the coproduct in $\vr V$
of two copies of ${\BZ}_2$.
Show that this coproduct is 
$D_4$, the eight element dihedral group.\footnote{Our
book uses the notation $D_8$ for this
group.}

\vfill

4. Let $\m r$ be a commutative ring.
Show that the tensor product of two finitely generated 
$\m r$-modules is again finitely generated.

\vfill

\noindent
{\bf Group 3.}

\medskip

5. (The Characterization of Products.) 
Recall that the composition of binary relations is defined by
$\beta\circ\gamma = \{(x,z)\;|\;\exists y\;((x,y)\in\beta 
\;\&\;(y,z)\in \gamma\}.$
Show that $\m a\cong\m b\times\m c$ if and only if 
$\m a$ has congruences $\beta$ and $\gamma$ such that
\begin{enumerate}
\item[(1)] $\beta\circ\gamma = 1$.
\item[(2)] $\beta\cap \gamma = 0$.
\item[(3)] $\m a/\beta \cong \m b$ and $\m a/\gamma\cong \m c$.
\end{enumerate}
($1$ is the largest congruence on $\m a$ and
$0$ is the smallest congruence on $\m a$.)

\vfill

6. Let ${\BH}$ be the quaternion algebra.
Show that ${\BH}\otimes_{\BR}{\BH}^{op}$ and 
$M_4({\BR})$ are isomorphic ${\BR}$-algebras.
(Hint: a typical member of $\BH$ has the form
$a + bi + cj + dk$, and so can be identified
with $[a\;b\;c\;d]\in {\BR}^4$. 
This identification of ${\BH}$ with ${\BR}^4$
turns ${\BR}^4$ into a left module over 
${\BH}\otimes_{\BR}{\BH}^{op}$, since 
${\BH}\otimes_{\BR}{\BH}^{op}$ acts on 
${\BH}$ according to $(g\otimes h)(v) = gvh$.
%$g, v, h\in {\BH}$.
This action of the ring ${\BH}\otimes_{\BR}{\BH}^{op}$ 
on the 4-dimensional space ${\BH}$ allows one to assign a 
$4\times 4$ real matrix $[w]$
to each $w\in {\BH}\otimes_{\BR}{\BH}^{op}$,
as in Homework III Additional Problems 3 \& 4.
Argue that $w\mapsto [w]$ is an isomorphism
from ${\BH}\otimes_{\BR}{\BH}^{op}$ to
$M_4({\BR})$.)

\vfill

\end{document}
