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



\begin{questions}

  \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 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 $p(t)$.


  \vspace{.5in}


  In the next couple of exercises, we explore the notions of prime and
  maximal ideals in non-commutative rings.  We start with two basic
  definitions.  Here $A$ is a general ring which does not necessarily
  have an identity.
  \begin{itemize}
  \item \textit{Definition:} A ring $A$ is \textbf{simple} if
    $A^2 \ne 0$, and its only (2-sided) ideals are $\{0\}$ and~$A$.
  \item \textit{Definition:} Let $P$ be an ideal in a ring $A$.  We
    say that $P$ is a \textbf{prime ideal} if $P \ne A$ and if for any
    ideals $I, J \subseteq A$, $IJ \subseteq P$ implies
    $I \subseteq P$ or $J \subseteq P$.
  \end{itemize}
  \question

  \begin{parts}

    \part Let $k$ be a field, and $n \ge 1$ an integer.  Show that the
    matrix ring $M_n(k)$ is a simple ring.  \textit{Hint:} Let
    $E_{i,j}$ be the $n\times n$ matrix with a $1_k$ in the
    $i,j$-entry, and $0_k$ elsewhere.  For a matrix $A \in M_n(k)$, it
    might be interesting to compute $E_{p,r} A E_{s,q}$.
  

    \part Let $A$ be a non-trivial ring with identity, $P \subseteq A$
    an ideal, $P \ne A$.
    \begin{itemize}
    \item Show that if for all $a, b \in A$, $ab \in P$ implies
      $a \in P$ or $b \in P$, then $P$ is a prime ideal (in the sense
      defined above).
    \item Show that if $A$ is commutative, and $P$ is prime in the
      above sense, then for all $a, b \in A$, we have $ab \in P$
      implies $a \in P$ or $b \in P$.  Where did you use
      commutativity?
    \item Find an example of an ideal in a non-commutative ring which
      is prime, but fails to satisfy the prime definition for
      commutative rings.
    \end{itemize}
    \part Let $A$ be a ring with identity. Show that an ideal
    $M \subseteq A$ is maximal if and only if $A/M$ is simple and
    nontrivial.  Show that every maximal ideal is prime.
  \end{parts}

  \question Let $\varphi: A \to B$ be a surjective homomorphism of
  (not necessarily commutative) rings with identity.
  \begin{parts}
    \part Let $Q \subset B$ be a prime ideal.  Show that
    $\varphi^{-1}(Q)$ is a prime ideal in $A$.

    \part Let $Q \subset B$ be a maximal ideal.  Show that
    $\varphi^{-1}(Q)$ is a maximal ideal in $A$. Give an example where
    $\varphi$ is not surjective, $\varphi^{-1}(Q) \ne A$, and yet
    $\varphi^{-1}(Q)$ is not maximal.


    \part Show that if $I$ is an ideal of $A$, then $\varphi(I) = B$
    if and only if $I + \ker(\varphi) =A $.  Conclude that if $M$ is a
    maximal ideal in $A$, that $\varphi(M)$ is maximal if and only if
    $\ker(\varphi) \subseteq M$.

  \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}

  

\end{questions}

\end{document}







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