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\title{Math 721\\Spring 2011\\Homework 2}
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\centerline{\it Due Friday February 4, 2011.}

\centerline{}

\noindent 1. ({\it The Five Lemma.}) Consider the following commutative diagram of $R-$module homomorphisms with exact rows
\begin{center}
$\xymatrix{A_1\ar[r]^{f_1}\ar[d]^{g_1} & A_2\ar[r]^{f_2}\ar[d]^{g_2} & A_3\ar[r]^{f_3}\ar[d]^{g_3} & A_4\ar[r]^{f_4}\ar[d]^{g_4} & A_5\ar[d]^{g_5}\\
B_1\ar[r]^{h_1} & B_2\ar[r]^{h_2} & B_3\ar[r]^{h_3} & B_4\ar[r]^{h_4} & B_5}$
\end{center}

\begin{itemize}
\item[a)] (5 pt) Show that if $g_2$ and $g_4$ are onto and $g_5$ is one to one then $g_3$ is onto.
\item[b)] (5 pt) Show that if $g_2$ and $g_4$ are one to one and $g_1$ is onto then $g_3$ is one to one.
\item[c)] (5 pt) Establish the short five lemma as a special case of a) and b).
\end{itemize}

\centerline{}

\noindent 2. ({\it The 3$\times$3 Lemma.})
Consider the following commutative diagram of $R-$module homomorphisms
\begin{center}
$\xymatrix{ & 0\ar[d] & 0\ar[d] & 0\ar[d] & \\
0\ar[r] & A_1\ar[r]\ar[d] & A_2\ar[r]\ar[d] & A_3\ar[r]\ar[d] & 0\\
0\ar[r] & B_1\ar[r]\ar[d] & B_2\ar[r]\ar[d] & B_3\ar[r]\ar[d] & 0\\
0\ar[r] & C_1\ar[r]\ar[d] & C_2\ar[r]\ar[d] & C_3\ar[r]\ar[d] & 0\\
 & 0 & 0 & 0 & } $
\end{center}

\begin{itemize}
\item[a)] (5 pt) Show that if the columns and the bottom two rows are exact, then the top row is exact.
\item[b)] (5 pt) Show that if the columns and the top two rows are exact, then the bottom row is exact.
\end{itemize}





\centerline{} 

\noindent 3. ({\it The Snake Lemma.}) Consider the following commutative diagram with exact rows
\begin{center}
$\xymatrix{ & A_1\ar[r]^{f_1}\ar[d]^{g_1} & A_2\ar[r]^{f_2}\ar[d]^{g_2} & A_3\ar[r]\ar[d]^{g_3} & 0 \\
0\ar[r] & B_1\ar[r]^{h_1} & B_2\ar[r]^{h_2} & B_3 & }$
\end{center}
\begin{itemize}
\item[a)] (5 pt) Show that there is an exact sequence
\begin{center}
$\xymatrix@1{\text{ker}(g_1)\ar[r]^{\alpha_1} & \text{ker}(g_2)\ar[r]^{\alpha_2} & \text{ker}(g_3)\ar[r]^{\partial} & \text{coker}(g_1)\ar[r]^{\beta_1} & \text{coker}(g_2)\ar[r]^{\beta_2} & \text{coker}(g_3)}$
\end{center} 
\item[b)] (5 pt) Show that if $f_1$ is one to one, then so is $\alpha_1$.
\item[c)] (5 pt) Show that if $h_2$ is onto, then so is $\beta_2$.
\end{itemize}


\centerline{}

\noindent 4. An $R-$module $S$ is said to be simple if the only submodules of $S$ are itself and $0$.
\begin{itemize}
\item[a)] (5 pt) Show that any simple $R-$module is cyclic (that is, is of the form $Ra$ for some $a\in S$).
\item[b)] (5 pt) Characterize all $R-$module homomorphisms $f:S\longrightarrow S$ where $S$ is a simple $R-$module.
\end{itemize}

\centerline{}

\noindent 5. Let $R$ be a ring, we define the {\it opposite ring}, $R^{\text{op}}$ to be the ring with the same underlying abelian group $(R,+)$ and multiplication given by $x*y=yx$ where $yx$ is ordinary multiplication.
\begin{itemize}
\item[a)] (5 pt) Show that $R^{\text{op}}$ is a ring.
\item[b)] (5 pt) Show that if $M$ is a left  $R-$module, then $M$ is a right $R^{\text{op}}-$module.
\end{itemize}
 



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