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\begin{document}



\title{Math 499/696, Spring 2010, Homework 5\\ Due Friday 09 April}


\maketitle

Item numbers refer to version of course notes dated March 29, 2010

\

\

\noindent\textbf{Exercise 3.5.26:}
Let $A$ be a commutative ring with identity, and let $R$ be the polynomial ring
$R=A[X_1,\ldots,X_d]$ in $d$ variables. 
Let $I$ and $J$ be monomial ideals in $R$.
\begin{enumerate}[(a)]
\item 
Prove that $\mrad I\subseteq\mrad J$ if and only if $\rad I \subseteq\rad J$.
\item 
Prove that $\mrad I=\mrad J$ if and only if $\rad I=\rad J$.
\end{enumerate}


\

\noindent\textbf{Exercise 3.5.27:}
Let $A$ be a commutative ring with identity and let $R$ be the polynomial ring
$R=A[X_1,\ldots,X_d]$ in $d$ variables. 
Set $\idealX=(X_1,\ldots,X_d)R$.  Prove that, if $I$ is a monomial ideal,
then $\mrad{I}=\idealX$ if and only if $\rad{I}=\rad{\idealX}$.



\

\noindent\textbf{Exercise 3.5.32:}
Let $A$ be a commutative ring with identity and let $R$ be the polynomial ring
$R=A[X_1,\ldots,X_d]$ in $d$ variables. 
Let $I_1,\ldots,I_n$ be monomial ideals in $R$ and set $\idealX=(X_1,\ldots,X_d)R$.  
\begin{enumerate}[(a)]
\item  \label{exer100306j2}
Prove that, if $\mrad{I_j}=\idealX$ for $j=1,\ldots,n$, then $\mrad{I_1\cdots I_n}= \idealX $.
\item  \label{exer100306j3}
Prove that if $I_j\neq R$ for $j=1,\ldots,n$ and
$\mrad{I_1\cdots I_n}= \idealX $, then $\mrad{I_j}=\idealX$ for $j=1,\ldots,n$.
\end{enumerate}


\

\noindent\textbf{Exercise 3.5.36:}
Let $A$ be a commutative ring with identity and let $R$ be the polynomial ring
$R=A[X_1,\ldots,X_d]$ in $d$ variables. 
Set $\idealX=(X_1,\ldots,X_d)R$, and let $I$ and $J$ be  monomial ideals of $R$.
Prove that if $I\subseteq\idealX$ and
$\mrad J=\idealX$, then $\cideal{J}{R}{I}\supsetneq J$.


\

\noindent\textbf{Exercise 3.5.31:}
[Math 696 only]
Let $A$ be a commutative ring with identity and let $R$ be the polynomial ring
$R=A[X_1,\ldots,X_d]$ in $d$ variables. 
Let  $\idealX=(X_1,\ldots,X_d)R$.  Let $I$ be a monomial ideal in $R$, and 
let $S$ denote the set of monomials in $R\ssm I$.
Prove that $S$ is a finite set if and only if
$\mrad{I}\supseteq\idealX$.






\end{document}














