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\noindent
\parbox{2.25in}{\bf Math 4720 \& 5210}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Mar. $25^\text{th}$, 2019}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 Prime and Maximal] As a quick reminder, in $\mathbb{Z}$ and in $\mathbb{Z}_n$, we know that subgroup = normal subgroup = cyclic subgroup = subring = ideal = principal ideal.
\begin{enumerate}[(a)]
\item Find all the ideals of $\mathbb{Z}_{24}$ and draw the corresponding lattice. Which ideals are prime? Which are maximal?
\item Determine which ideals in $\mathbb{Z}$ are prime and which are maximal. [Prove your assertions.]

{\it Note:} Don't forget to consider the trivial ideal: $\{0\}$.
\end{enumerate}

\item[\#2 Another Ideal Problem] Let $R$ be a ring and let $I$ and $J$ be ideals of $R$.
\begin{enumerate}[(a)]
\item Show that $I+J = \{ i+j \;|\; i \in I \text{ and } j \in J\}$, $I \cap J$, and $IJ = \{ i_1j_1+\cdots+i_mj_m\;|\; m\geq 0; i_k \in I \text{ and } j_k \in J\}$ are ideals of $R$.
\item Let $R=\mathbb{Z}$, $I=(9)$, and $J=(12)$. What are $I+J$, $I \cap J$, and $IJ$?
\item Let $R$ be a {\bf commutative} ring with $1$. We say $I$ and $J$ are coprime (or relative prime or comaximal) if $I+J=R$. Suppose that $I+J=R$ (i.e., $I$ and $J$ are relatively prime).
         Show that $I\cap J=IJ$ and $\diagquotient{R}{IJ}=\diagquotient{R}{I \cap J} \cong \diagquotient{R}{I} \times \diagquotient{R}{J}$.
         
         {\it Note:} This is called Chinese Remaindering. 
         
         {\it Hints:} $IJ \subseteq I \cap J$ is always true. The reverse containment requires the ``coprime'' assumption. For the next statement, use the first isomorphism theorem. 
          Also, there exists $i \in I$ and $j \in J$ such that $i+j=1$ (Why?). This decomposition of $1$ will help in proving both statements. Finally, for the isomorphism part, 
          consider $aj+bi$. How is this related to $a$ mod $I$? $b$ mod $J$?
\end{enumerate}

\item[\#3 Idealistic Divisibility] Let $R$ be an integral domain. Recall that $a$ divides $b$ iff $b$ is a multiple of $a$ iff there is some $k \in R$ such that $ak=b$ iff $b\in (a)$ iff $(b) \subseteq (a)$.
\begin{enumerate}[(a)]
\item Let $a,b\in R$. We say $d\in R$ is a {\it greatest common divisor} (GCD) of $a$ and $b$ iff $d$ is a common divisor of $a$ and $b$ (i.e., $d$ divides $a$ and $d$ divides $b$) and also given any other common divisor $c$ (i.e., $c$ divides $a$ and $c$ divides $b$) we have that $c$ divides $d$. 

Suppose that $(a)+(b)=(a,b) = \{ax+by \;|\; x,y \in R\}$ is principal, say $(a,b)=(d)$. Show that $d$ is a GCD of $a$ and $b$.

\item {\bf [Grad. Students]} Give a similar definition for a {\it least common multiple} (LCM) of $a$ and $b$. Show that if $(a)\cap (b) = (\ell)$, then $\ell$ is an LCM of $a$ and $b$.
\end{enumerate}

\item[\#4 Fractionally Important] {\bf [Grad. Students]} Let $R$ be a principal ideal domain (PID) and let $S$ be a multiplicative subset of $R$ (i.e., $a,b \in S$ implies $ab \in S$) and also assume that $0 \not\in S$. Show that $RS^{-1}$ is also a PID. [Recall that $RS^{-1} = \{ r/s \;|; r \in R \text{ and } s \in S \}$ is the ring of fractions with numerators in $R$ and denominators in $S$.]

{\it Hint:} Let $\mathcal{I}$ be an ideal of $RS^{-1}$. Consider $I = \{ a \in R \;|\; \text{there exists some } s \in S \text{ such that } a/s \in \mathcal{I} \}$ (i.e. the set of numerators). 
Show $I$ is an ideal of $R$ and $IS^{-1} = \{ a/s \;|\; a \in I \text{ and } s\in S\} = \mathcal{I}$.


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