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\begin{center}
  {\large Mathematics 101}\\Fall 2014\\Homework 2
\end{center}


\begin{enumerate}

\item Irreducible and Indecomposable Modules
  \begin{enumerate}
 
  \item Show that if an $R$-module $M$ is irreducible, then it is
    cyclic, that is $M = Rm$ for some $m \in M$.  Characterize all
    irreducible $\Z$-modules.
  \item Show that $\Q$ is an indecomposable $\Z$-module.
  \item Let $V = k^2$ be a two dimensional vector space over a field
    $k$, and let $T$ be a linear operator on $V$ so that with respect
    to some basis, the matrix of $T$ has the form: $\begin{pmatrix}
      1&1\\0&1
    \end{pmatrix}$.  Viewing $V$ as a $k[x]$-module (induced by $T$),
    show that $V$ is reducible, but indecomposable.
  \end{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.  An $R$-module $M$ is finitely
  generated if there is a finite subset $\{m_1, \dots, m_t\}$ of $M$
  so that every element of $M$ can be written as an $R$-linear
  combination of the $m_i$.

  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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