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\title{進階代數(上)第二次作業}
\date{2008~年九月二十二日}
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\author{上課老師: 翁志文 }

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\pointname{分}

\begin{document}

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%       這份試卷有 \   \numquestions\ 大題,\ 共計\  \numpoints\
%  分.\  請詳細提供所有過程!}}}
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We always assume that a ring $R$ has the multiplication identity $1.$

\bigskip

\begin{questions}
\question (林家銘) Let $A$, $B$ be ideals of a ring $R.$ Show that $AB$ is an ideal of $R,$ where
$$AB:=\{a_1b_1+a_2b_2+\cdots+a_nb_n~|~a_i\in A, b_i\in B\}.$$

\question (林育生) Let $R$ be a commutative ring and $P$ be an ideal of $R.$ Show that $P$ is prime
if and only if the quotient ring $R/P$ is an integral domain.

\question   Let $M$ be an ideal of a ring $R.$  $M$ is said to be {\it maximal} if $M\not= R$ and
for any ideals $N$, we have
$$M\subseteq N\subseteq R\Longrightarrow N=M~{\rm or~}N=R.$$
\begin{parts}
\part (陳建文) Show that if $M$ is a maximal ideal and $R$ is commutative then the quotient ring $R/M$ is a
field.
\part (羅健峰)   Show that if the quotient ring $R/M$ is a division ring, then $M$ is maximal.
\part  (何昕暘)  Give an example of a maximal ideal $M$ in a ring $R$ such that $R/M$ is not a division ring.
\end{parts}


\question

\begin{parts}
\part (賴德展 ) Let $I_1$, $I_2,$ $I_3$  be ideals of a ring $R$ such that $I_i+(I_j\cap I_k)=R$ for $\{i, j,
k\}=\{1, 2, 3\}.$ Show that for any $a_1, a_2, a_3\in R$ there exists $a\in R$ such that $a-a_i\in
I_i$ for all $i\in\{1, 2, 3\}.$ (Hint. For each $a_i$ there exists $b_i\in I_i$ and $c_i\in I_j\cap
I_k$ such that $b_i+c_i=a_i.$)

\part  (洪湧昇) Let $I_1$, $I_2,$ $I_3$  be ideals of a ring $R$ such that $I_i+(I_j\cap I_k)=R$ for $\{i, j,
k\}=\{1, 2, 3\}.$ Show that $R/(I_1\cap I_2\cap I_3)$ is isomorphic to direct product $R/I_1\times
R/I_2\times R/I_3:=\{(\overline{a_1}, \overline{a_2}, \overline{a_3})~|~\overline{a_i}\in R/I_i\}$
of rings $R/I_1$, $R/I_2$ and $R/I_3$ with addition and multiplication are defined componentwise.

\part  (林志峰) Let $n_1, n_2, n_3$ be positive integers  such that any two of them are relative primes. Show that
for any integers $a_1, a_2, a_3$ there exists an integer $a$ such that $a\equiv a_i\pmod {n_i}$ for
$i=1, 2, 3.$ (這是中國餘式定理)

\part (呂融昇) Let $n_1, n_2, n_3$ be positive integers  such that any two of them are relative primes. Show
that $\mathbb{Z}/(lcm(n_1, n_2, n_3))$ is isomorphic to $\mathbb{Z}/(n_1)\times
\mathbb{Z}/(n_2)\times \mathbb{Z}/(n_3).$
\end{parts}

\question An ideal $J$ is {\it radical} of $R$ if $a^n\in J$ for some positive integer $n$ implies
$a\in J.$ Let $I$ be an ideal of a commutative ring $R$ and set
$$\sqrt{I}:=\{a\in R~|~a^n\in I~{\rm for~some~positive~integer~}n\}.$$
\begin{parts}
\part (連敏筠)   Find $\sqrt{(24)}$ in $\mathbb{Z}.$
\part (施智懷)  Show that $\sqrt{I}$ is a radical ideal of $R.$
\end{parts}

\end{questions}

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