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\title{Math 724\\Fall 2010\\Homework 6}
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\centerline{\it Due Monday, May 10, 2010}

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\noindent 1. Let $R$ be a very strongly atomic ring and $e\in R$ an idempotent. 
\begin{itemize}
\item[a)] (5 pt) Show that $e$ is either $0$ or $1$. 
\item[b)] (5 pt) Use this to find all values of $n>1$ such that $\mathbb{Z}/n\mathbb{Z}$ is very strongly atomic.
\end{itemize}

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\noindent 2. Let $R$ be an atomic domain and $x\in R$ a nonzero, nonunit. Let Irr($R$) denote the set of distinct (up to associates) irreducible divisors of $x$. We define the irreducible divisor graph of the element $x$ ($G_x$) to be the graph with vertices from Irr($R$) and we declare that if $u,v\in\text{Irr}(R)$ we say that $u$ and $v$ have an edge between them if $uv$ divides $x$ (we ignore the possibility of loops). Prove the following conditions are equivalent.
\begin{itemize}
\item[a)] $R$ is a UFD.
\item[b)] $G_x$ is connected for all nonzero, nonunits $x\in R$.
\item[c)] $G_x$ is complete for all nonzero, nonunits $x\in R$.
\end{itemize}

  
 





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