%        File: hw1.tex
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\begin{document}
\begin{center}
    \textsc{Math 4140/5140. Homework 10}\\
    due Wednesday, April 20
\end{center}

\vspace{1em}

\noindent\textbf{Note:} All numbered exercises are from Erdmann--Holm ([EH]).\\

\noindent 
Let $k$ be a field.

\begin{enumerate}
    \item Read Chapter 5 of [EH]. 
    \item Show that for any $k$-algebras $A_1,A_2,\dots, A_r$, we have \[
            (A_1\times A_2\times \cdots \times
        A_r)^{\op}\cong A_1^{\op}\times A_2^{\op}\times \cdots \times A_r^{\op}\]
        as algebras.
    \item Prove that for any division algebra $D$ over $k$, the transpose map \[
            M_n(D^{\op})\ra (M_n(D))^{\op},\quad
        X\mapsto X^T\]
        is an
     algebra isomorphism.
 %\item In the proof of Lemma 5.6, the authors show that the map $\Phi$ is bijective by checking injectivity and
     %surjectivity separately. In class, we discussed the surjectivity proof. Complete either one of the following problems
      %to
     %finish the proof that $\Phi$ is bijective:
     %\begin{enumerate}
         %\item Show that $\Phi$ is injective.
         %\item Show that $\Phi$ is bijective by finding a two-sided inverse map.
     %\end{enumerate}
     %Please use the same notation as the book/our notes. (You don't need to recall the meaning of the notation.) 
 \item Exercise 5.1. 
 \item Exercise 5.2. (\emph{Hint}: Proposition 1.29 and
     Example 1.30 may help.) 
 \item Exercise 5.9. 

\end{enumerate} 
\end{document}


