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\pagestyle{fancy}
\lhead{MATH 522--01}
\chead{Problem Set \#1\answer{ solutions}}
\rhead{}% Your name here!
\lfoot{}
\cfoot{\answer{Page \thepage\ of \pageref{LastPage}}}
\rfoot{due Thursday, January 22, 2015}

\begin{document}
\begin{enumerate}
\item \question{Let $U$ be a nonempty set and let $R$ be the set of
    all subsets of $U$ (i.e. the power set of $U$). For the two given
    proposed definitions of ``addition'' and ``multiplication'',
    determine whether $R$ is a ring or not; if it is not a ring,
    explain why, and if it is a ring, identify its identity elements.}
  \begin{enumerate}
  \item \question{$A+B=A\cup B$ and $A\cdot B=A\cap B$.}

    \answer{}


  \item \question{$A+B=(A\cup B)-(A\cap B)$ (also known as the
      ``symmetric difference'') and $A\cdot B=A\cap B$.}

    \answer{}

  \end{enumerate}

\item \question{Prove that for a commutative ring $R$ and $x\in R$,
    $x$ is a unit of $R$ if and only if $x$ divides every element of
    $R$.}
  
  \answer{}

\item \question{Prove the following two statements:}
  \begin{enumerate}
  \item \question{For any ring $R$ and a (not necessarily finite)
      collection $\mathcal R$ of subrings of $R$, their intersection
      $$\bigcap_{S\in\mathcal R} S$$
      is also a subring of $R$. (Hint: you may find it easier to warm
      up by addressing just an intersection of two subrings of $R$)}

    \answer{}

  \item \question{For a ring $R$ and a subset $S$ of $R$, there is a
      unique \emph{smallest} subring $\langle S\rangle$ of $R$ which
      contains all the elements of $S$ (hint: choose an appropriate
      collection $\mathcal R$ for the above statement).}

    \answer{}
  \end{enumerate}

\item \question{Address the two following questions about subrings:}
  \begin{enumerate}
  \item \question{Let $\mathbb Q[\sqrt 2,\sqrt 3]$ denote the
      smallest subring of $\mathbb R$ which contains every rational
      number, the irrational number $\sqrt 2$, and the irrational
      number $\sqrt 3$. Let $\mathbb Q[\sqrt 2+\sqrt 3]$ denote the
      smallest subring of $\mathbb R$ which contains every rational
      number and the irrational number $\sqrt 2+\sqrt 3$ (note that
      the previous question guarantees that these structures are
      well-defined). Prove that $\mathbb Q[\sqrt 2,\sqrt 3]=\mathbb
      Q[\sqrt 2+\sqrt 3]$.}

    \answer{}
    
  \item \question{Let $\mathbb Z[\sqrt 2,\sqrt 3]$ and $\mathbb
      Z[\sqrt 2+\sqrt 3]$ be defined as in the previous part except
      with ``rational number'' replaced by ``integer'' in the
      definition. Demonstrate that $\mathbb Z[\sqrt 2,\sqrt
      3]\neq\mathbb Z[\sqrt 2+\sqrt 3]$.}

    \answer{}
  \end{enumerate}

\item \question{Prove that if the additive group of a ring $R$ is
    cyclic, then $R$ is commutative.}

  \answer{}

\end{enumerate}
\end{document}