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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due January 27 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4130/6130 $|$ Visual Algebra $|$
    Spring 2023 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}

\begin{enumerate}

  %%--------------------------------------------------------------------

\item Let $R$ be a commutative ring with $1$.
  \begin{enumerate}
  \item Show that the following two conditions are equivalent:
    \begin{enumerate}
    \item[(i)] the non-units of $R$ form an ideal,
    \item[(ii)] $R$ has a unique maximal ideal.
    \end{enumerate}
  \item Characterize the units of the following ring
    \[
    S=\big\{a/b\mid a,b\in\Z,\;\gcd(a,b)=1,\,\text{$b$ is odd}\big\},
    \]
    and show that the nonunits form a unique maximal ideal. \medskip
  \end{enumerate}

  
\item Let $R$ be a commutative ring with $1$, and $D\subseteq R$ a
  multiplicatively closed subset containing no zero divisors. Consider the
  following set and equivalence relation $\sim$:
  \[
  R\times D=\big\{(r,d)\mid r\in R,\,d\in D\big\},\qquad (r_1,d_1)\sim
  (r_2,d_2) \;\Leftrightarrow\; r_1d_2=r_2d_1.
  \]
  \begin{enumerate}
  \item Show that $\sim$ is an equivalence relation.
  \item Let $r/d$ denote the equivalence class containing $(r,d)$, and
    the set of equivalence classes by $D^{-1}R$. Define addition and
    subtraction as follows:
    \[
    \frac{r_1}{d_1}+\frac{r_2}{d_2}:=\frac{r_1d_2+r_2d_1}{d_1d_2}\qquad\text{and}
    \qquad
    \frac{r_1}{d_1}\times\frac{r_2}{d_2}:=\frac{r_1r_2}{d_1d_2}.
    \]
    Show that these operations are well-defined.
  \item Show that the additive and multiplictive identities are $0/d$
    and $d/d$, for any $d\in D$, and that the multiplicative inverse of $r/d$, if it exists, is $(r/d)^{-1}=d/r$.
  \item If $d\in D$, show that $\{rd/d\mid r\in R\}$ is a subring of
    $D^{-1}R$ and that
    \[
    R\longto D^{-1}R,\qquad r\longmapsto rd/d
    \]
    is a injective homomorphism, therefy identifying $R$ with a
    subring of $D^{-1}R$.
  \item Under this identification, show that every $d\in D$ gets
    mapped to a unit in $D^{-1}R$.
  \item Show how the ring $S$ from the previous problem
    arises as ring $D^{-1}R$ of fractions, for some $D$ and $R$. \medskip
  \end{enumerate}

  %%----------------------------------------------------------------
  
  \item Let $D$ be a multiplicatively closed subset of a commutative
    ring $R$, and $f\colon R\into S$ an embedding to a commutative
    ring with $1$, such that $f(d)$ is a unit, for every $d\in
    D$. Show that there is a unique ring homomorphism $h\colon
    D^{-1}R\into S$ such that $h\circ \iota=f$.
    \[
    \tikzstyle{e} = [draw, -stealth']
    \begin{tikzpicture}[scale=3.5]
      \begin{scope}[shift={(0,0)}]
        \node (G) at (0,0) {$R$};
        \node (A) at (1,0) {$S$};
        \node (G/G') at (.5,-.5) {$D^{-1}R$};
        \draw [right hook->] (G) -- (A) node[midway,above]{\footnotesize $f$};
        \draw [right hook->] (G) -- (G/G') node[midway,below left]{\footnotesize $\iota$};
        \draw [right hook->,dashed] (G/G') -- (A) node[midway,below right]{\footnotesize $h$};
      \end{scope}
      %%
      \begin{scope}[shift={(1.75,0)}]
        \node (G) at (0,0) {$r$};
        \node (A) at (1,0) {$s$};
        \node (G/G') at (.5,-.5) {$r/1$};
        \draw [e,|->] (G) -- (A) node[midway,above]{\footnotesize $f$};
        \draw [e,|->] (G) -- (G/G') node[midway,below left]{\footnotesize $\iota$};
        \draw [e,|->] (G/G') -- (A) node[midway,below right]{\footnotesize $h$};
      \end{scope}
    \end{tikzpicture}
    \]
  

\end{enumerate}
  
  %%----------------------------------------------------------
  
\end{document}


