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\begin{center}
  {\large Mathematics 101}\\Fall 2014\\Homework 5
\end{center}

\begin{enumerate}
\item Let $R$ be a PID and $M$ a finitely generated $R$-module.  Show
  that $M$ is projective if and only if it is free.

\item Let $M$ be a submodule of $\Z^n$ having group index $p$ in $M$,
  i.e., $[\Z^n:M] = p$, where $p$ is a prime.  Show that $M$ is free
  of rank $n$, and there is a basis $\{e_1, \dots, e_n\}$ of $\Z^n$ so
  that $M = \Z e_1 \oplus \cdots \oplus \Z e_{n-1} \oplus \Z pe_n$.

\item Show that a vector $v=(a_1, \dots, a_n) \in \Z^n$ extends to a
  basis $\{v, v_2, \dots, v_n\}$ of $\Z^n$ if and only if the $a_i$
  are coprime, that is $a_1\Z + \cdots + a_n\Z = \Z$.  \textit{Hint:}
  For one direction, come up with a short exact sequence that splits.

\item Let $F_1$ and $F_2$ be free modules of (not necessarily the
  same) finite rank over a PID $R$.  Let $\varphi: F_1 \to F_2$ be
  $R$-linear and nontrivial.  Show that there exists bases $\{v_1,
  \dots, v_n\}$ of $F_1$ and $\{w_1, \dots, w_m\}$ of $F_2$ together
  with elements $a_1, \dots, a_r$ of $R$ so that
  \[
  \varphi(v_i) =
  \begin{cases}
    a_iw_i&1 \le i \le r\\0&r+1\le i\le n,
  \end{cases}
  \]
  with $a_1\mid a_2\mid \dots\mid a_r$, and the ideals $a_jR$ uniquely
  determined.
\item Let $A =
  \begin{pmatrix}
    4&7&2\\2&4&6
  \end{pmatrix}$.

  \begin{enumerate}
  \item If $\varphi: \Z^3 \to \Z^2$ is a $\Z$-linear map whose matrix
    with respect to the standard bases is $A$, determine the structure
    of the cokernel $\Z^2/Im(\varphi)$ as a direct sum of cyclic
    groups.  Find a minimal set of generators for the quotient.
    \textit{Hint:} The image of $\varphi$ is the span of the columns
    (i.e., the column space), and you may assume without loss of
    generality that elementary column operations (over $\Z$) leave the
    column space unchanged.  Explain how your answer is connected to
    the invariant factor theorem.

  \item Determine all integer solutions to $A
    \begin{pmatrix}
      x\\y\\z
    \end{pmatrix}
    = 0$.  \textit{Hint:} Elementary row operations (over $\Z$) do not
    change the kernel.
  \end{enumerate}

\end{enumerate}






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