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\title{Algebra Homework Set 7}
\author{James Cummings (jcumming@andrew.cmu.edu)}


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   Due by class time on Wednesday 2 November. Homework must be typeset and submitted by email as a PDF file. 

\begin{enumerate}

\item Recall that a subset $S$ of a ring $R$ is multiplicatively closed (MC) if $1 \in S$ and
  $S$ is closed under multiplication. Let $S \subseteq R$ be MC and define a binary relation
  $\sim$ on $R \times S$ as follows: $(r, s) \sim (r', s')$ iff there is $t \in S$ such that
  $t(r s' - r' s) = 0$.  Prove that:
  \begin{enumerate}
  \item  $\sim$ is an equivalence relation.
  \item  Defining $+$ and $\times$ as in the definition of field of fractions makes the
    set of $\sim$-classes into a ring (which we write $R S^{-1}$). Just check the operations
    are well-defined, it is then clear that the ring axioms are satisfied. 
  \end{enumerate}

\item Let $G$ be a torsion-free $\mathbb Z$-module (abelian group) of rank $1$ and let $P$ be the set of prime numbers.
\begin{enumerate}
\item Prove that $G$ is isomorphic to a subgroup of $(\mathbb Q, +)$.
\item Let $G \le {\mathbb Q}$. For each nonzero $a \in G$, let $n_a: P \rightarrow {\mathbb N} \cup \{\infty\}$
  be defined as follows:
  \[
  n_a(p) = \sup \{ n: a/p^n \in G \}.
  \]
  Prove that if $a$ and $b$ are both nonzero then $n_a(p) = \infty \iff n_b(p) = \infty$, and
  $\{ p : n_a(p) \neq n_b(p) \}$ is finite.  
\end{enumerate} 
  
   
\item Let $A$ be an $n \times n$ integer matrix and let $G_A$ be the subgroup of ${\mathbb Z}^n$ generated  
  by the columns of $A$. Prove that ${\mathbb Z}^n/G_A$ is finite iff $det(A) \neq 0$, and that in
  this case ${\mathbb Z}^n/G_A$ has order $\vert det(A) \vert$.
  
\item Prove that the intersection of any nonempty chain of prime ideals is prime. 

\item Let $R$ be a PID and let $N$ be a free $R$-module on a countably infinite set of generators
  (for example the set of all functions from ${\mathbb N}$ to $R$ which are zero on a cofinite set).
  Prove that every submodule of $N$ is free.
  
  
\end{enumerate}
  
  
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