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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 3.3  \\ Fall 2019 \hfill Due September 18}}}

\large

\vspace{1cm}

\begin{exercise}
Let $I = (X)$ be the principal ideal generated by $X$ in $\Z[X]$.  Prove that for any $n \ge 2$ there is a strictly increasing chain
\[
I = I_1 \subsetneqq I_2 \subsetneqq \cdots \subsetneqq I_n
\]
of ideals in $\Z[X]$ ({\em cf.\ }exercise 3.1.3).
\end{exercise}

\begin{exercise}
Show that multiplication of ideals is associative.
\end{exercise}

\begin{exercise}
Let $A$ and $B$ be ideals in a ring $R$.  For $n \in \N$ let
\[
\begin{aligned}
nA &= \underbrace{A + A + \cdots + A}_{n \text{ summands}},\\
A^n &= \underbrace{AA \cdots A}_{n \text{ factors}}.
\end{aligned}
\]

\begin{itemize}

\item[\bf a.]  Prove that $nA = A$ for all $n \ge 1$.

\item[\bf b.]  Show that $\cdots \subset A^3 \subset A^2 \subset A$.

\item[\bf c.]  If $A+B = R$, prove that $A^n + B^n = R$ for all $n \ge 1$.

\end{itemize}

\end{exercise}

\begin{exercise}
Let $R$ be a commutative principal ideal ring in which $A^2 = A$ for every ideal $A$.  Prove that every nonzero element of $R$ is either a unit or a zero divisor (compare to exercise 1.2.4).  With a good deal more work one can show that, in fact, any such ring is isomorphic to a product of finitely many fields. 
\end{exercise}

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