\documentclass[12pt]{article}
\usepackage[psamsfonts]{amsfonts}
\usepackage{amsmath, amsthm, amssymb}
\usepackage{fullpage}
\newcommand{\ds}{\displaystyle}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Zp}{\Z_{(p)}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\cyclic}[1]{\la #1 \ra}
\DeclareMathOperator{\Tor}{Tor}


\begin{document}
\baselineskip=18pt \thispagestyle{empty}

\begin{center}
  {\large\textbf{Dartmouth College}\\} Mathematics 101\\\medskip
  Homework 6 \small{(due Wednesday, November 14)}
\end{center}

\begin{enumerate}

  
\item If $R$ is a ring with identity, an element $e\in R$ is called an
  \textit{idempotent} if $e^2=e$.  Notice that aside from $e=0,1$, all
  other idempotents are zero divisors.  If $\varphi: R\to S$ is a
  homomorphism between rings with identity, $\varphi(1_R)$ is an
  idempotent, so to find homomorphisms in which $\varphi(1_R)$ is not
  the identity, one must look for idempotents in $S$.  Find all
  \textit{ring} homomorphisms $\varphi: \Z_{120} \to \Z_{42}$; verify
  they are ring homomorphisms.

\item Let $R$ be a commutative ring with identity.  An element $x\in
  R$ is called \textit{nilpotent}, if $x^n=0$ for some positive
  integer $n$.
  \begin{enumerate}
  \item Show that the set of nilpotent elements in $R$ form an ideal,
    called the \textit{nilradical} of $R$.
  \item Show that the sum of a unit and a nilpotent element is a unit
    in $R$. \textit{Hint:} As a lemma, show it first with the unit
    equal to the identity.


  \end{enumerate}
\item Let $R$ be a commutative ring with identity. Let $R[x]$ be the
  polynomial ring in one variable with coefficients in $R$.  Let $p(x)
  = a_0 + a_1 x+ \cdots + a_n x^n \in R[x]$.

  \begin{enumerate}
  \item Show that $p$ is nilpotent in $R[x]$ if and only if $a_i$ is
    nilpotent in $R$ for all $i\ge 0$.

  \item Show that $p$ is a unit in $R[x]$ if and only if $a_0$ is a
    unit in $R$ and the $a_i$ are nilpotent in $R$ for $i\ge 1$.
    \textit{Hint:} If $p= a_0 + \cdots + a_n x^n$ and $q = b_0 +
    \cdots + b_m x^m$ with $pq = 1$, show by induction on $r$ that
    $a_n^{r+1}b_{m-r} = 0$ to conclude $a_n$ nilpotent for $n\ge 1$.
  \end{enumerate}

\item Assume that $R$ is a commutative ring with identity, and let
  $f(x)$ be a monic polynomial in $R[x]$ of degree $n\ge 1$.  Let bar
  notation denote passage to the quotient ring $R[x]/(f)$.
  \begin{enumerate}
  \item Show that every element of $R[x]/(f)$ has a unique
    representative of the form $\overline{a_0 + a_1x + \cdots +a_{n-1}
      x^{n-1}}$ with $a_i \in R$.
  \item If $f(x) = x^n -a$ for $a$ some nilpotent element of $R$, show
    that $\overline x$ is nilpotent in $R[x]/(f)$.
  \end{enumerate}
\end{enumerate}

\end{document}






