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\title{Math 724\\Fall 2005\\Homework 3}
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\centerline{\it Due Wednesday October 19, 2005.}

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\noindent 1. (5 pt) Show that $R$ is a PID if and only if $R$ is a UFD of dimension no more than 1.

\centerline{}

\noindent 2. (5 pt) Let $R$ be a commutative ring with identity and $T$ an extension of $R$ that is (finitely) generated by $m$ elements over $R$. If the Krull dimension of $R$ is $n$, show that 

\[
\text{dim}(T)\leq 2^mn+2^m-1.
\]

\noindent What is the lower bound for $\text{dim}(T)$?

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\noindent 3. Recall that $V$ is a valuation domain if given two nonzero elements $a,b\in V$ then either $a$ divides $b$ or $b$ divides $a$. A Bezout domain is a domain where every finitely generated ideal is principal and a GCD domain is a domain where every two (nonzero) elements have a greatest common divisor.

\begin{itemize}
\item[a)] (5 pt) Show that any we have the implications 
\[
\text{Valuation domain}\Longrightarrow\text{Bezout domain}\Longrightarrow\text{GCD domain}.
\]
\item[b)] (5 pt) Show that none of the implications above are reversible.
\item[c)] (5 pt) Show that $R$ is a valuation domain if and only if it is Bezout and quasi-local.
\item[d)] (5 pt) Show that $R$ is a Bezout domain if and only if it is a GCD domain with the property that $\text{gcd}(a,b)$ is a linear combination of $a$ and $b$.
\item[e)] (5 pt) Show that any GCD domain (and hence Bezout and valuation) is integrally closed.
\end{itemize}



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