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\pagestyle{fancy} \lhead{{\sf Homework 14 $|$ Due December 3 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2021 $|$ M.~Macauley}}

\begin{document}

%$\;$ \vspace{-4mm}

\begin{enumerate}

  %%--------------------------------------------------------------------
  
\item Let $R$ be a commutative ring with $1$.
  \begin{enumerate}
  \item Show that $R$ is an integral domain if and only if $0$ is a
    prime ideal.
  \item Show that an ideal $P\subseteq R$ is prime if and only if
    $R/P$ is an integral domain.
  \item Show that every maximal ideal is prime. 
  \item Find the group of units $U(R)$ and the maximal ideal(s) of the ring
    \[
    R=\Big\{\frac{a}{b}:a,b\in\Z,\;
    \gcd(a,b)=1,\;p\nmid b\Big\},
    \]
    where $p$ is a fixed prime number. \smallskip
  \end{enumerate}

  %%--------------------------------------------------------------------
  
\item An ideal $I\subseteq R$ is \emph{radical} if $x^n\in I$
  implies that $x\in I$. It is \emph{primary} if $ab\in I$ implies
  that either $a\in I$ or $b^n\in I$ for some $n\in\N$. An element $r\in
  R$ is \emph{nilpotent} if $r^n$ for some $n\in\N$.
  \begin{enumerate}
  \item Show that $I\subsetneq R$ is radical if and only if $R/I$
    has no nonzero nilpotent elements.
  \item Show that the following are equivalent for $I\subsetneq R$:
    \begin{enumerate}[(i)]
    \item $I$ is prime
    \item $I$ is radical and primary
    \item The ideal
      \[
      I[x]:=\left\{\Sum_{k=0}^n a_kx^k\mid a_k\in I\right\}
      \]
      is a prime ideal of $R[x]$.
    \end{enumerate}
  \item Let $R$ be a principal ideal domain. Characterize all
    nonzero proper ideals that are radical, and all nonzero proper
    ideals that are primary. \smallskip
  \end{enumerate}
  
  %%--------------------------------------------------------------------
  
\item Let $R$ be a principal ideal domain. A \emph{common
  multiple} of $a,b\in R^*$ is an element $m$ such that $a\mid m$ and
  $b\mid m$. Moreover, $m$ is a \emph{least common multiple} (lcm) if
  $m\mid n$ for any other common multiple $n$ of $a$ and $b$.
  \begin{enumerate}
  \item Show that any $a,b\in R^*$ have an lcm.
  \item Show that an lcm of $a$ and $b$ is unique up to multiplication
    of associates, and can be characterized as a generator of the
    (principal) ideal $I:=(a)\cap (b)$. \smallskip
  \end{enumerate}

  %%--------------------------------------------------------------------
  
\item For any squarefree $m\in \Z$, the \emph{ring of quadradic
  integers} $R_m$ is the subring
  \[
  R_m:=\big\{a+b\sqrt{m}\mid a,b\in\Z\big\}\;\text{ if $m\equiv_4 2,3$},
  \quad\quad
  R_m:=\big\{\tfrac{a}{2}+\tfrac{b}{2}\sqrt{m}\mid a,b\in\Z\big\}\;\text{ if $m\equiv_4 1$}
    \]
  of the field $\Q(\sqrt{m})=\big\{r+s\sqrt{m}\mid r,s\in\Q\big\}$.
  For any $x=r+s\sqrt{m}\in\Q(\sqrt{m})$, define the \emph{norm} of
  $x$ to be $N(x)=r^2-ms^2$.
  \begin{enumerate}
  \item Show that $N(xy)=N(x)N(y)$.
  \item Show that $N(x)\in\Z$ if $x\in R_m$.
  \item Show that $u\in U(R_m)$ if and only if $|N(u)|=1$. 
  \item Show that $U(R_{-1})=\{\pm 1,\pm i\}$, $U(R_{-3})=\{\pm
    1,\pm(1\pm\sqrt{-3})/2\}$, and $U(R_m)=\{\pm 1\}$ for all other
    negative square-free $m\in\Z$. %\bigskip
  \end{enumerate}
  
  
\end{enumerate}

\end{document}



