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\begin{document}

\noindent Math 742, Spring 2016\\
Homework 4\\
Due: February 19

\section{Exercises}

\begin{enumerate}[(1)]

\item Let $R$ be an integral domain and let $R[x]$ be the polynomial ring in one variable. Given $f \in R[x]$, the {\bf content} of $f$, denoted ${\rm cont}(f)$, is the ideal of $R$ generated by the coefficients of $f$.
  \begin{enumerate}
  \item Given $f,g \in R[x]$, show that
\[
{\rm cont}(fg) \subseteq {\rm cont}(f) {\rm cont}(g) \subseteq \sqrt{{\rm cont}(fg)}.
\]
\item Pick $a,b,c,d \in R[x]$ and assume that $ab = cd$. Show that if $p \in R$ is a prime element that divides $a$, then $p$ divides either $c$ or $d$.

\item Now assume $R$ is a unique factorization domain. Prove {\bf Gauss' Lemma}: Let $K$ be the fraction field of $R$. Show that if $f \in R[x]$ is irreducible, then $f$ is also irreducible in the larger ring $K[x]$.
  \end{enumerate}

\bigskip

\item Let $n$ be a square-free integer (i.e., every prime divides $n$ at most once). Let $\bQ$ be the rational numbers. Define
  \begin{align*}
    \bQ(\sqrt{n}) &= \{a + b\sqrt{n} \mid a,b \in \bQ\}.
  \end{align*}
  \begin{enumerate}
  \item Verify that $\bQ(\sqrt{n})$ is a field.
  \item If $b \ne 0$, show that $a+b\sqrt{n}$ satisfies a unique monic degree 2 polynomial with rational coefficients.
  \item Determine the integral closure of $\bZ$ in $\bQ(\sqrt{n})$. 
  \item What happens if we don't assume $n$ is square-free?
  \end{enumerate}

\bigskip

\item Let $M$ be an $R$-module. Define the {\bf support} of $M$ to be
\[
\Supp(M) = \{P \in \Spec(R) \mid M_P \ne 0\}.
\]
\begin{enumerate}
\item Show that $\Supp(M) \subseteq V(\Ann(M))$, and that equality holds if $M$ is finitely generated.
\item Give an example where $\Supp(M)$ is not a closed subset of $\Spec(R)$ (and in particular is not equal to $V(\Ann(M))$).
\item Let $N$ be another $R$-module. Show that $\Supp(M \otimes_R N) \subseteq \Supp(M) \cap \Supp(N)$, and that equality holds if $M$ and $N$ are finitely generated.
\end{enumerate}

\bigskip

\item Let $0 \to A \to B \to C \to 0$ be a sequence of $R$-modules. Show that the following are equivalent:
  \begin{enumerate}
  \item $0 \to A \to B \to C \to 0$ is exact.
  \item $0 \to A_P \to B_P \to C_P \to 0$ is exact for all prime ideals $P$.
  \item $0 \to A_P \to B_P \to C_P \to 0$ is exact for all maximal ideals $P$.
  \end{enumerate}

\bigskip

\item In this exercise, we'll explore localization in the noncommutative setting. So, in this exercise, $R$ denotes a not necessarily commutative ring, i.e., we have all of the axioms for a ring except $ab=ba$ is no longer required.
  \begin{enumerate}
  \item Given a multiplicative subset $S \subseteq R$, call a ring homomorphism $f \colon R \to R'$ {\bf $S$-inverting} if $f(s)$ is a unit for all $s \in S$. Show that there exists a ring $R_S$, along with an $S$-inverting map $\phi \colon R \to R_S$ which is universal in the sense that for any other $S$-inverting map $f \colon R \to R'$, there exists a unique $g \colon R_S \to R'$ such that $f = g \circ \phi$. If $R$ is commutative, show that $R_S = S^{-1} R$.

  \item In the commutative setting, we can construct $R_S$ using ``fractions'', but this might not be possible in general: let $k$ be a (commutative) field and let $R = k\langle X,Y \rangle$ be the ring of noncommutative polynomials\footnote{A noncommutative monomial is a sequence of $X$'s and $Y$'s and we take the product by concatenating them, e.g., $(X^3 Y X^2 Y^5)(Y X^2) = X^3 Y X^2 Y^6 X^2$. $X$ and $Y$ do not commute, but they do commute with the elements of $k$, and a noncommutative polynomial is a finite linear combination of noncommutative monomials with coefficients in $k$.}. Describe the ring $R_S$ where $S$ is the multiplicative subset generated by $\{X,Y\}$.

  \item Exercise 11.2 of Altman--Kleiman says that, if $R$ is commutative, then $S^{-1} R = 0$ if and only if $S$ contains a nilpotent element. This can also fail in the noncommutative setting: Let $k$ be a field and let $n \ge 2$ be an integer. Set $R = M_n(k)$ to be the ring of $n \times n$ matrices with entries in $k$. For $1 \le i,j \le n$, let $E_{ij}$ be the matrix\footnote{These are called matrix units.} with a $1$ in the $(i,j)$ position and 0's elsewhere. Show that $R_S = 0$ where $S = \{E_{1,1}\}$.

  \end{enumerate}

\end{enumerate}


\section{Suggested exercises (don't submit)}

From Altman--Kleiman:
\begin{itemize}
\item Chapter 10: 22, 35
\item Chapter 11: 2, 8, 18, 25, 32
\item Chapter 12: 6, 8, 14, 28
\end{itemize}

\section{Further reading}

The issues that come up in Exercise 5 illustrate that localization for noncommutative rings can be subtle. See Chapter 4 of T.-Y. Lam, {\it Lectures on Modules and Rings} for more information on noncommutative localization.

\end{document}
