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\title{進階代數(上)第十一次作業}
\date{2008~年十二月十一日}
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\author{上課老師: 翁志文 }

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\begin{document}

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%       這份試卷有 \   \numquestions\ 大題,\ 共計\  \numpoints\
%  分.\  請詳細提供所有過程!}}}
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\begin{questions}

\question (葉彬) Determine all the $n\times n$ matrices in rational canonical form over
$\mathbb{R}$ with minimal polynomial $\lambda^2+1.$

\question (The Five Lemma) Suppose we have the following commutative diagram of $R$-module
homomorphisms with exact rows.
$$\left.
  \begin{array}{ccccccccc}
   ~~~~~M_1 &  \longrightarrow & ~~~~~M_2 & \longrightarrow  & ~~~~~M_3  & \longrightarrow  & ~~~~~M_4 & \longrightarrow  & ~~~~~M_5\\
     & & & & & & && \\
    f_1\displaystyle\downarrow &  \displaystyle\circlearrowright & f_2\displaystyle\downarrow & \displaystyle \circlearrowright & f_3\displaystyle\downarrow &  \displaystyle\circlearrowright& f_4\displaystyle\downarrow & \displaystyle\circlearrowright&  f_5\displaystyle\downarrow  \\
            & & & & & & && \\
    ~~~~~N_1 & \longrightarrow & ~~~~~N_2 & \longrightarrow & ~~~~~N_3& \longrightarrow & ~~~~~N_4 & \longrightarrow & ~~~~~N_5 \\
  \end{array}
\right.$$

\begin{parts}
\part (林家銘) Prove directly $f_1$ surjection and $f_2$, $f_4$ injection $\Longrightarrow$ $f_3$ injection.
\part (林育生) Prove (a) by using The Short Five Lemma.
\part (陳健文) Prove directly $f_5$ injection and $f_2$, $f_4$ surjection $\Longrightarrow$ $f_3$ surjection.
\part (羅健鋒) Prove (c) by using  The Short Five Lemma.
\end{parts}

\question Let $P_i$, $i\in I,$ be $R$-modules.
\begin{parts}
\part (何昕暘) Show that if $\oplus_{i\in I}P_i$ is projective then $P_i$ is projective for each $i\in
I.$
\part (賴德展) Show that if  $P_i$ is projective for each $i\in
I$ then $\oplus_{i\in I}P_i$ is projective.
\end{parts}



\end{questions}

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