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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 3.1  \\ Fall 2019 \hfill Due September 18}}}

\large

\vspace{1cm}

\begin{exercise}
Let $I \subset J$ be ideals in a ring $R$.  Prove that the rule $a+I \mapsto a+J$ yields a well-defined homomorphism $R/I \to R/J$, and that the diagram
\[
\xymatrix{R \ar[d] \ar[dr] &  \\  R/I \ar[r] & R/J }
\]
commutes, where the vertical and diagonal arrows are the canonical surjections.
\end{exercise}

\begin{exercise}
Let $I$ be an ideal in a commutative ring $R$.  The {\em radical} of $I$ is the set 
\[
\sqrt{I} = \{ a \in R \, | \, a^n \in I \text{ for some } n \in \N \},
\]
which consists of all the elements in $R$ that are $n$th roots of some element in $I$, for all $n \ge 1$.  

\begin{itemize}

\item[\bf a.]  Prove that $\sqrt{I}$ is an ideal in $R$ containing $I$.

\item[\bf b.]  Prove that $\sqrt{\sqrt{I}} = \sqrt{I}$.  Conclude that $R/\sqrt{I}$ has no nonzero nilpotent elements.

\item[\bf c.]  Which elements of $R$ are in $\sqrt{0}$?

\end{itemize}

\end{exercise}

\begin{exercise}
Let $I$ be a nonzero proper ideal in $\Z$.  Prove that there is a constant $C(I)$ so that if $I=I_1 \subset I_2 \subset \cdots \subset I_n$ is a strictly increasing sequence of proper ideals in $\Z$, then $n \le C(I)$.
\end{exercise}



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