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\begin{center} {\large \textbf{Math 721 -- Homework 7}} \medskip \\ Due Friday, March 6 at 5pm  \\ \ \\ 
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\noindent Good practice problems (do not turn in solutions): \\
DF 15.1: 1, 6, 7, 8 10, 11, 29, 30   \\
DF 15.2: 2, 7, 30, 31, 32, 34, 38\\


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\begin{prob}[DF 15.1 Exercise 2]
Show that each of the following rings are \emph{not} Noetherian by exhibiting an explicit infinite increasing chain of ideals.  \smallskip 
\begin{itemize}
\item[(a)] the ring of continuous real valued functions on $[0,1]$,  \smallskip 
\item[(b)] the ring of all functions from $\N$ to $\Z/2\Z$. 
\end{itemize} 
 \end{prob}
 
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\begin{prob}[DF 15.2 Exercises 39,40] Let $R$ be a Noetherian ring and suppose 
\[I = \cap_{i=1}^mQ_i\]
is a minimal primary decomposition of an ideal $I\subset R$. For each $i$, let $P_i = \rad(Q_i)$ be the prime associated to $Q_i$. 
For $a\in R$, define 
\[I:\langle a \rangle = \{ r\in R : ar \in I\}.\]
\begin{itemize}
\item[(a)] Show that $I:\langle a \rangle$ is an ideal of $R$ and $I:\langle a \rangle= R$ if and only if $a\in I$. \smallskip 
\item[(b)] Show that for any ideals $I$ and $J$, $(I \cap J):\langle a \rangle = (I:\langle a\rangle) \cap  (J:\langle a\rangle) $.  \smallskip 
\item[(c)] Show that if $a\not\in Q_i$, then $Q_i:\langle a \rangle$ is primary with $\rad(Q_i:\langle a \rangle) = P_i$ and that if  
$a\not\in P_i$, then $Q_i:\langle a \rangle=Q_i$.  \smallskip 
\item[(d)] Show that 
\[ I:\langle a \rangle = \cap_{i=1}^m (Q_i : \langle a \rangle) \text{ and }  \rad(I:\langle a) \rangle = \cap_{i=1}^m \rad(Q_i : \langle a \rangle) .\]
\item[(e)] Show that $\rad(I:\langle a\rangle )$ is the intersection of the primes $P_i$ for which $a\not\in Q_i$.  \smallskip 
\item[(f)] Show that if $\rad(I:\langle a\rangle )$ is prime, then $\rad(I:\langle a\rangle) = P_i$ for some $i$.  \smallskip 
\item[(g)] Show that for each $i=1, \hdots, m$, there exists an element $a\in R$ with $\rad(I: \langle a \rangle) = P_i$.\\ (Hint: consider $a\in ( \cap_{j\neq i}Q_j) \backslash Q_i$.)  \smallskip 
\item[(h)] Show that for a prime ideal $P\subseteq R$, $P$ has the form $\rad(I:\langle a\rangle)$ for some $a\in R$ if and only if $P=P_i$ for some $i$. 
\end{itemize}
This shows that the set of primes associated to an ideal $I$ is unique.
\end{prob} 

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