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\title{Math 720\\Fall 2010\\Homework 7}
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\centerline{\it Due Friday, November 19, 2010.}

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\noindent 1. (5 pt) Show that any finite commutative ring with no nonzero zero divisors is a field.


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\noindent 2. We say that the commutative ring with identity, $R$, is zero-dimensional if every prime ideal is maximal.
\begin{itemize}
\item[a)] (5 pt) Show that any zero-dimensional domain is a field.
\item[b)] (5 pt) Show that if $R$ is finite, then $R$ is zero dimensional.
\item[c)] (5 pt) Give an example of an infinite zero-dimensional ring that is not a field. 
\end{itemize}
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\noindent 3. An element $e\in R$ is said to be a idempotent if $e^2=e$ and $e$ is said to be a central idempotent if $e$ is idempotent and $e\in Z(R)$.
\begin{itemize}
\item[a)] (5 pt) Show that if $R$ has an identity and $e$ is a central idempotent, then $1-e$ is also a central idempotent.
\item[b)] (5 pt) If $R$ has an indentity, show that $eR$ and $(1-e)R$ are ideals of $R$ such that $R\cong eR\times(1-e)R$.
\item[c)] (5 pt) If $R$ has a central idempotent, does it follow that $R$ has an identity?
\end{itemize}
  



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\noindent 4. Let $R$ be a principal ideal ring (PIR) with identity (you may assume that such a ring is commutative with identity).
\begin{itemize}
\item[a)] (5 pt) Show that if $R$ is an integral domain then either $R$ is a field or $R$ is one-dimensional (meaning that every nonzero prime ideal is maximal).
\item[b)] (5 pt) Show that the homomorphic image of a PIR is a PIR.
\item[c)] (5 pt) Show that if $\mathfrak{P}_1\subsetneq\mathfrak{P}_2$ are prime ideals of $R$, then $\mathfrak{P}_2$ is maximal (i.e. $R$ is no more than 1-dimensional).
\item[d)] (5 pt) Is it true that any subring of a PIR is a PIR?
\end{itemize}




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