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\begin{center}
  {\large Mathematics 111}\\Spring 2009\\Homework 2
\end{center}


\begin{enumerate}

\item Let $R$ be a ring with identity.  Show that the sequence of
left $R$-modules
\[\xymatrix{%
  0 \ar@{>}[r]& L \ar@{>}[r]^\varphi &M \ar@{>}[r]^\psi&N}
\]
is exact if and only if for all left $R$-modules $D$, the sequence
\[\xymatrix{0 \ar@{>}[r]&\Hom_R(D,L) \ar@{>}[r]^{\varphi_*}&
  \Hom_R(D,M) \ar@{>}[r]^{\psi_*}& \Hom_R(D,N)}\]
is exact.



\medskip
\noindent\textit{Hint:} We have done the forward direction in class;
for the converse, a single propitious choice of $D$ can work, but you
still need to sweat the details.


\item Let $R$ be a ring with identity.  Show that the sequence of
left $R$-modules
\[\xymatrix{%
  L \ar@{>}[r]^\varphi &M
 \ar@{>}[r]^\psi&N  \ar@{>}[r] &0 }
\]
is exact if and only if for all left $R$-modules $D$, the sequence
\[\xymatrix{0 \ar@{>}[r]&\Hom_R(N,D) \ar@{>}[r]^{\psi^*}&
  \Hom_R(M,D) \ar@{>}[r]^{\varphi^*}& \Hom_R(L,D)}\]
is exact.



\medskip
\noindent\textit{Hint:} We have done the forward direction in class.
The converse is more complicated than the covariant version; you may
want to choose different modules $D$ to establish the various
conditions determining exactness of the original sequence.  For
example, to show $\psi$ is surjective, let $D = N/Im(\psi)$ (the
cokernel of $\psi$), and $\pi: N \to D$ the natural projection. Now
consider $\psi^*(\pi)$ and its implications.

As a second hint, to show $Im(\varphi) \subseteq Ker(\psi)$, you need
only show that $\psi\circ \varphi = 0$.  Choose $D = N$ and consider
the identity map $id_N \in \Hom_R(N,D) = \Hom_R(N,N)$.


\item Let $R$ be a ring with identity and consider the short exact
  sequence of $R$-modules:
\[\xymatrix{%
  0 \ar@{>}[r]& L \ar@{>}[r]^\varphi &M \ar@{>}[r]^\psi&N\ar@{>}[r]&0}
\]
\begin{enumerate}
\item Show that if $L$ and $N$ are finitely generated, so is $M$.
\item Show that if $M$ is finitely generated, so is $N$.
\item Show by example that if $M$ is finitely generated, $L$ need not be.
\end{enumerate}

\item Determine the number of group homomorphisms $\Z_{12} \oplus
  \Z_{14} \to \Z_{20}$, and explicitly characterize them by
  specifying their action on $(\bar 1, \bar 0)$ and $(\bar 0, \bar
  1)$.

\end{enumerate}


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