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\title{Algebra Homework Set 6}
\author{James Cummings (jcumming@andrew.cmu.edu)}


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   Due by class time on Wednesday 19 October. Homework must be typeset and submitted by email as a PDF file. 

\begin{enumerate}

\item  Prove that if $r$ is a nilpotent element in a ring $R$ then $r$ is in every prime ideal of $R$.

\item  Prove that if $r$ is in every prime ideal of $R$ then $r$ is nilpotent. Hint: Apply a question from
  last week to show that if $r$ is not nilpotent then it lies in at least one prime ideal.

  
\item  Let $R$ be the subring of the complex numbers consisting of elements of the form
  $m + n \alpha$ where $m, n \in {\mathbb Z}$ and  $\alpha = i \sqrt{5}$.
  Define $N(m + n \alpha) = m^2 + 5 n^2$.

  
\begin{enumerate}

\item  Prove that $N(r) = 0$ iff $r = 0$, and $N(r s) = N(r) N(s)$.

\item  Identify the units of $R$..

\item   Prove that $2$, $3$, $1 + \alpha$ and $1 - \alpha$ are irreducible in $R$.

\item Prove that none of the elements $2$, $3$, $1 + \alpha$ and $1 - \alpha$ are prime in $R$.
  Hint: $2 \times 3 = (1 - \alpha)(1 + \alpha)$.

\item   Prove that $R$ is not a PID. 
  
\end{enumerate}  


\item  Let $R$ be a subring of $S$, let $a \in S$ and define $R[a]$ to be the least subring
  of $S$ containing $R \cup \{ a \}$.

  \begin{enumerate}

  \item  Prove that $R[a] = \{ f(a) : f \in R[x] \}$.

  \item  Prove that $R[a]$ is fg as an $R$-module iff there is a monic $g \in R[x]$ such that
    $g(a) = 0$.

  \end{enumerate}
  
    
\item  Let $R$ be a ring. Prove that the sum of a unit and a nilpotent element is a unit.

\item  Let $R$ be a ring and let $f = c_0 + c_1 x + \ldots c_n x^n \in R[x]$.
  Prove that the following are equivalent:
\begin{enumerate}
\item $f$ is a unit of $R[x]$.
\item $c_0$ is a unit and $c_i$ is nilpotent for all $i > 0$.
\end{enumerate} 
  Hint: You may find the preceding question and the fact (from last HW) that
  nilpotent elements form an ideal helpful.

  
\end{enumerate}
  
  
\end{document} 
