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\name{}
\class{Math 171 - Abstract Algebra I, Section \uline{\ \ \ }}
\assignment{HW \#13}
\duedate{10/25/10}

\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}

\begin{document}

\problemlist{Dummit \& Foote (7.4) 24, 30, 31, 32}

Remember, $R$ is a ring with identity $1 \neq 0$.

\begin{problem}[7.4.24]
Prove that in a Boolean ring every finitely generated ideal is principal.
\end{problem}
\newpage

\begin{problem}[7.4.30]
Let $I$ be an ideal of the commutative ring $R$ with identity and define
$$ \text{rad}\ I = \{ r \in R\ |\ r^n\in I \text{ for some }n \in \Z^+\}$$
called the {\it radical} of $I$.  Prove that rad $I$ is an ideal containing $I$ and that (rad $I)/I$ is the nilradical of the quotient ring $R/I$, i.e., (rad $I)/I = \mathfrak{N}(R/I)$ (cf. Exercise 29, Section 3).
\end{problem}
\newpage

\begin{problem}[7.4.31]
An ideal $I$ of the commutative ring $R$ with identity is called a {\it radical ideal} if rad $I = I$.
\begin{itemize}
\item[(a)] Prove that every prime ideal of $R$ is a radical ideal.
\item[(b)] Let $n > 1$ be an integer.  Prove that 0 is a radical ideal in $\Z/n\Z$ if and only if $n$ is a product of distinct primes to the first power (i.e., $n$ is square free).  Deduce that $(n)$ is a radical ideal of $\Z$ if and only if $n$ is a product of distinct primes in $\Z$.
\end{itemize}

\end{problem}
\newpage

\begin{problem}[7.4.32]
Let $I$ be an ideal of the commutative ring $R$ with identity and define
\begin{center} Jac $I$ to be the intersection of all maximal ideals of $R$ that contain $I$\end{center}
where the convention is that Jac $R = R$.  (If $I$ is the zero ideal, Jac 0 is called the {\it Jacobson radical} of the ring $R$, so Jac $I$ is the preimage in $R$ of the Jacobsen radical of $R/I$.)
\begin{itemize}
\item[(a)] Prove that Jac $I$ is an ideal of $R$ containing $I$.
\item[(b)] Prove that rad $I \subseteq $ Jac $I$, where rad $I$ is the radical of $I$ defined in Exercise 30.
\item[(c)] Let $n > 1$ be an integer.  Describe Jac $n\Z$ in terms of the prime factorization of $n$.
\end{itemize}
\end{problem}

\end{document} 