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\thispagestyle{empty}
\begin{flushright}
  Your Name Here
\end{flushright}




\begin{center}
  \textbf{Math 71}\\ Homework 7
\end{center}

All rings are assumed to have identity with $1\ne 0$.

\begin{itemize}
\item [p249. \#24] Let $\varphi: R\to S$ be a ring homomorphism.
  
  \begin{enumerate}[label=(\alph*)]

  \item Prove that if $J$ is an ideal of $S$, then $\varphi^{-1}(J)$
    is an ideal of $R$.  Apply this to the special case where $R$ is a
    subring of $S$ and $\varphi$ is the inclusion homomorphism to
    deduce that if $J$ is an ideal of $S$, then $J\cap R$ is an ideal
    of $R$.

  \item Prove that if $\varphi$ is surjective and $I$ is an ideal of
    $R$, then $\varphi(I)$ is an ideal of $S$.  Give an example where
    this fails if $\varphi$ is not injective.
  \end{enumerate}

\item [p250. \#26c] Prove that if $p$ is a prime, and $R$ is a
  commutative ring with characteristic $p$, then $(a+b)^p = a^p + b^p$
  for all $a, b \in R$.

\item [p250. \#34] Let $I$, $J$ be ideals of a ring $R$.

  \begin{enumerate}[label=(\alph*)]
  \item Prove that $I+J$ is the smallest ideal containing $I$ and $J$.
  \item Prove that $IJ$ is an ideal contained in $I\cap J$.

  \item Give an example where $IJ \ne I\cap J$.
  \item Prove that if $R$ is commutative and $I+J=R$, then
    $IJ = I\cap J$.
  \end{enumerate}

\item [p 257. \#10] Assume $R$ is commutative.  Prove that if $P$ is a
  prime ideal of $R$ and $P$ contains no zero divisors, then $R$ is an
  integral domain.

\item[p257. \#11] Assume $R$ is commutative.  Let $I$ and $J$ be
  ideals of $R$, and assume that $P$ is a prime ideal of $R$ that
  contains $IJ$.  Prove that either $I$ or $J$ is contained in $P$.

\item [p257. \#14] Assume $R$ is a commutative ring.  Let $x$ be an
  indeterminate, $f(x)$ a monic polynomial in $R[x]$ of degree
  $n \ge 1$.  Use the bar notation $g \mapsto \overline {g}$ to denote
  passage from
  \[g(x) \in R[x] \mbox{ to } \overline{ g(x)} \in R[x]/(f(x)).
  \]
          
           
  \begin{enumerate}[label=(\alph*)]
  \item Show that every element of $R[x]/(f(x))$ is of the form
    $\overline{p(x)}$ for some polynomial $p(x) \in R[x]$ of degree
    less than $n$.  Hint: if $f(x) = x^n + b_{n-1}x^{n-1}+\cdots+b_0$,
    then $\overline{x^n} = \overline{-( b_{n-1}x^{n-1}+\cdots+b_0)}$.
    Use this to reduce powers of $\overline x$ in the quotient ring.

  \item Prove that if $p(x)$ and $q(x)$ are distinct polynomials in
    $R[x]$ which are both of degree less than $n$, then
    $\overline{p(x)} \ne \overline{q(x)}$.

  \item If $f(x) = a(x)b(x)$ with both $a(x)$ and $b(x)$ having degree
    less than $n$, prove that $\overline{a(x)}$ is a zero divisor in
    $R[x]/(f(x))$.
  \item If $f(x) = x^n-a$ for some nilpotent element $a \in R$, prove
    that $\overline{x}$ is nilpotent in $R[x]/(f(x))$.
  \item Let $p$ be a prime and assume that $R = \F_p$ (the field with
    $p$ elements), and $f(x) = x^p-a$ for some $a \in R$.  Prove that
    $\overline{x-a}$ is nilpotent in $R[x]/(f(x))$ (cf. Exercise 26c
    above)
  \end{enumerate}
             
\end{itemize}
 

\end{document}


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