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\begin{document}

\textbf{Math 6310, Assignment 5 \hfill Due: 9th November, Monday}

% \medskip {\large Name: }

\begin{enumerate}

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\item If $1-ab$ is invertible in a ring, show that $1-ba$ is also invertible.

%\textbf{Solution:}

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\item Let $R$ be a ring in which $x^3=x$ for each $x$. Prove that $R$ is commutative.

%\textbf{Solution:}

\medskip
\textbf{In the problems below, $R$ is a commutative ring.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item In which of the following rings is every ideal principal? Justify your answer.
\[
(\text{a}) \ \ZZ \times\ZZ, \qquad 
(\text{b}) \ \ZZ/4, \qquad
(\text{c}) \ (\ZZ/6)[x], \qquad
(\text{d}) \ (\ZZ/4)[x].
\]
%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item If $R$ is a domain that is not a field, prove that the polynomial ring $R[x]$ is not a principal ideal domain.

%\textbf{Solution:}

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\item An element $r$ in a ring $R$ is \emph{nilpotent} if $r^{n}=0$ for some $n\ge 0$. Prove the following assertions.
\begin{enumerate}
\item If $r$ is nilpotent, then $1+r$ is invertible in $R$.
\item If $r_{1},\dots,r_{c}$ are nilpotent elements, then any element in the ideal $(r_{1},\dots,r_{c})$ is nilpotent.
\end{enumerate}

%\textbf{Solution:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $R$ be a commutative ring and $R[x]$ the polynomial ring over $R$ in the indeterminate $x$. Let 
\[
f(x)=r_{0}+r_{1}x+r_{2}x^{2}+ \cdots + r_{n}x^{n}\quad\text{with $r_{i}\in R$.}
\]
Prove the following assertions.
\begin{enumerate}
\item  $f(x)$ is nilpotent if and only if $r_{0},\dots,r_{n}$ are nilpotent.
\item $f(x)$ is a unit in $R[x]$ if and only if $r_{0}$ is a unit in $R$ and $r_{1},\dots,r_{n}$ are nilpotent.
\item $f(x)$ is a zerodivisor if and only if there exists a nonzero element $r\in R$ such that $r\cdot f=0$.
\end{enumerate}
Recall that $a\in R$ is a \emph{zerodivisor} if there exists $b\neq 0$ in $R$ with $ab=0$; the only zerodivisor in a domain is $0$.

%\textbf{Solution:}

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

