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\title{Math 420-620\\Fall 2012\\Homework 13}
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\centerline{\it Due Friday November 30, 2012.}

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\noindent 1. Let $R$ be a commutative ring with identity and let $I\subseteq R$ be an ideal.
\begin{itemize}
\item[a)] (5 pt) Show that $I$ is a maximal ideal if and only if $R/I$ is a field.
\item[b)] (5 pt) Show that $I$ is a prime ideal if and only if $R/I$ is an integral domain.
\item[c)] (5 pt) Show that $I$ is a radical ideal if and only if $R/I$ is a reduced ring.
\end{itemize}

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\noindent 2. Let $R$ be an integral domain. We say that the nonunit $\pi\in R$ is irreducible if $\pi=ab$ implies that either $a$ or $b$ is a unit. We say that the nonunit $0\neq p\in R$ is a (nonzero) prime element if $p$ divides $ab$ implies that $p$ divides $a$ or $p$ divides $b$.
\begin{itemize}
\item[a)] (5 pt) Show that $p\in R$ is prime if and only if $(p)$ is a prime ideal.
\item[b)] (5 pt) Show that any nonzero prime element is irreducible.
\item[c)] (5 pt) Give an example of an integral domain, $R$, and an element $\pi\in R$ that is irreducible, but not prime.
\item[d)] (5 pt) Show that if $a$ can be factored into a product of primes, then this factorization is unique (up to ordering and units).
\end{itemize}

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\noindent 3. Let $R$ be a PID (principal ideal domain).
\begin{itemize}
\item[a)] (5 pt) Show that every nonzero prime ideal is maximal.
\item[b)] (5 pt) Show that $R$ satisfies the ascending chain condition on principal ideals; that is, if you have the chain of principal ideals

\[
(a_1)\subseteq (a_2)\subseteq (a_3)\subseteq\cdots
\]
\noindent then there is an $n$ such that $(a_{n+k})=(a_n)$ for all $k\geq 0$.
\item[c)] (5 pt) Show that every nonzero nonunit element of $R$ can be factored uniquely (up to ordering and units) into prime elements. 
\end{itemize}





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