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\centerline{\textbf{Dartmouth College}} \centerline{Mathematics 81/111
  --- Homework 2}

\begin{questions}

\question The following is a very standard result whose proof you can
  easily find, but of course the value to you is in coming up with
  your own proof.

  Let $A$ be a commutative ring.  Show that the following conditions
  defining a Noetherian ring are equivalent:
  \begin{enumerate}[(a)]
  \item Every ideal of $A$ is finitely generated.
  \item (ACC) If $I_1 \subseteq I_2 \subseteq \cdots $ is an ascending
    chain of ideals in $A$, then there exists an integer $r \ge 1$ so
    that $I_r=I_{r+1} = \cdots $.
  \item Every nonempty collections of ideals in $A$ has a maximal
    element with respect to inclusion.
  \end{enumerate}


\question Let $\Q$ be the rational numbers.

  \begin{parts}
  \part Let $A$ be the ring $\Q\times \Q$.  Determine all the ideals
    of $A$, and which among those are maximal.


  \part In case we have not yet proved it in class, assume that the
    polynomial ring $\Q[x]$ is a PID.  Use the Chinese Remainder
    Theorem (as needed) to help characterize the structure of the
    quotient rings $\Q[x]/(x^2 - 3x +2)$ and $\Q[x]/(x^2 + x + 1)$.
    In particular, where in the spectrum of `commutative rings to
    fields' do these rings lie?

  \part Find the maximal ideas of the rings $\Q[x]/(x^2 - 3x +2)$ and
    $\Q[x]/(x^2 + x + 1)$.


  \end{parts}


\question Consider the ring $A = \Z[\alpha]$ where
  $\alpha = \root3\of 2$. Let $I = (5,\alpha+2)$ and
  $J = (5, \alpha^2 + 3\alpha -1)$ be two ideals of $A$.
  \begin{parts}
  \part Show that the principal ideal $5A = IJ$.
  \part Show that there is a surjective ring homomorphism: \newline
    $\Z[x]/(5, x^2+3x-1) \to \Z[\alpha]/(5, \alpha^2+3\alpha-1)$.
  \part Use the previous part to show that $(5, \alpha^2+3\alpha-1)$
    is a maximal ideal in $A=\Z[\alpha]$.
  
  \end{parts}

\question Let $A$ be a commutative ring with identity.

  \begin{parts}
  \part Show that if the polynomial ring $A[x]$ is a PID, then $A$ is
    a field (hence $A[x]$ is a Euclidean domain).

  \part Suppose that $A$ is an integral domain containing an
    irreducible element $\pi$.  Show that $A[x]$ is not a PID.  Hint:
    Consider the ideal $\la x,\pi \ra$.

  \part Hilbert's Basis theorem says that if $A$ is a Noetherian
    commutative ring with identity, then so is the polynomial ring
    $A[x]$.  Determine whether the converse is true.

  \end{parts}
  \vfill \rightline{(problems continue on next page)}

\question Let $\varphi: \C[x,y] \to \C[t]$ be the ring homomorphism
  between polynomial rings induced by sending $x \mapsto t^2$ and
  $y \mapsto t^3$.  Show that the kernel of $\varphi$ is the ideal
  $(y^2 - x^3)$ and that the image of $\varphi$ is the subring
  \[\{p(t) \in \C[t] \mid p'(0) = 0\},\] where $p'(t)$ is the first
  derivative of the polynomial $p(t)$.

\end{questions}

\end{document}







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