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\begin{center}
{\large\textbf{Dartmouth College}\\}
Mathematics 101\\\medskip Homework 6 \small{(due Thursday, Nov 12)}
\end{center}

\begin{enumerate}
\item In a ring $R$ without identity, one can still define the
  characteristic as zero or the least positive integer $n$ such that
  $na = 0_R$ for all $a \in R$.

  Show that every ring $R$ without identity can be embedded in a ring
  $S$ with identity in at least two ways; one in which $S$ has
  characteristic zero and the other where $S$ has the same
  characteristic as $R$.

  For the characteristic zero model, let $S = R \oplus \Z$ be the
  additive abelian group, and define a multiplication by ($r_1, k_1)
  (r_2, k_2) = (r_1r_2 + k_2r_1 + k_1 r_2, k_1k_2)$.  Verify that $S$
  is a ring with identity $(0,1)$ and characteristic 0, and that the
  map $R \to S$ given by $r \mapsto (r,0)$ is an injective ring
  homomorphism.

  When the characteristic of $R$ is $n > 0$, do a similar construction
  with $S = R \oplus \Z/n\Z$.


\item Let $R$ be a commutative ring with identity, and let $\N(R) =
  \{x \in R \mid x \textrm{ is nilpotent}\}$; nilpotent means that
  $x^n = 0$ for some positive integer $n$.
  \begin{enumerate}
  \item $\N(R)$ is called the nilradical of $R$; show that $\N(R)$ is
    an ideal of $R$.
  \item Show that $\N(R) = \cap_\P \P$ where $\P$ ranges over all the
    prime ideals of $R$.  Hint: The inclusion $\N(R) \subseteq \cap_\P
    \P$ is quite straightforward.  For the other consider the
    contrapositive.  Let $x\in R$ which is not nilpotent, and let $S$
    be the set of ideals $I$ in $R$ with the property that $x^n \notin
    I$ for $n \ge 1$.  Show that Zorn's lemma applies to $S$ and that
    a maximal element is a prime ideal not containing the element $x$.

 
  \end{enumerate}

\item If $\varphi: R \to S$ is a surjective homomorphism of rings,
  show that there is a one-to-one correspondence between the prime
  ideals of $S$ and the prime ideals of $R$ which contain
  $ker(\varphi)$.


\item Characterize the prime ideals of $R = \Z/n\Z$ ($n > 1$), and
  determine its nilradical, $\N(R)$.

\item For $R$ a commutative ring with identity, and $I \subseteq R$ an
  ideal, define the \textit{radical} of $I$, rad$(I) = \sqrt I$, to be
  $\{ r \in R \mid r^n \in I \textrm{ for some }n\ge 1\}$.  An ideal
  $I$ is called a \textit{radical ideal} if $\sqrt I = I$.
  \begin{enumerate}
  \item Show that $\sqrt I$ is an ideal containing $I$
  \item Show that every prime ideal is radical.
  \item Show that $\sqrt I/I = \N(R/I)$.
  \item Show that $\sqrt I$ is the intersection of all prime ideals of
    $R$ which contain $I$.
  \end{enumerate}

\item The Jacobson radical of a ring $R$ with identity is the
  intersection of all the maximal (left) ideals of $R$.  More
  generally, we can define the Jacobson radical of an ideal $I$,
  Jac$(I)$, to be the intersection of all maximal left ideals which
  contain $I$; then the Jacobson radical of $R$ is simply Jac$(0)$.
  Below we assume $R$ is commutative.
  \begin{enumerate}
  \item Show that the Jacobson radical of $R$ contains the nilradical.
  \item Show that for an ideal $I$, $\sqrt I \subset Jac(I)$.
  \item Compute the Jacobson radical of $\Z/n\Z$, $n \ge 2$.
  \end{enumerate}
\end{enumerate}


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