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\title{Math 420-620\\Fall 2012\\Homework 12}
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\centerline{\it Due Wednesday November 21, 2012.}

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\noindent 1. Let $R$ be a commutative ring with identity and let $x,y\in R$ be nilpotent elements.
\begin{itemize}
\item[a)] (5 pt) Show that $x+y$ and $xy$ are nilpotent elements.
\item[b)] (5 pt) Show that if $u$ is a unit of $R$ and $x$ is nilpotent, then $u+x$ is a unit.
\item[c)] (5 pt) Show that if $R$ is not commutative, neither of the above necessarily holds ($x+y$ is not necessarily nilpotent and $u+x$ is not necessarily a unit).
\end{itemize}

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\noindent 2. Let $R$ be a finite commutative ring.
\begin{itemize}
\item[a)] (5 pt) Show that if $R$ contains an element that is not a zero divisor, then $R$ has an identity.
\item[b)] (5 pt) Explain why every element of $R$ is either a zero divisor or a unit.
\item[c)] (5 pt) Show that any finite integral domain is a field.
\end{itemize}

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