\documentclass{amsart}
\usepackage{amssymb}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\fa}{\mathfrak{a}}
\newcommand{\fb}{\mathfrak{b}}

\begin{document}
\textbf{Suggested problems week 1}
\begin{enumerate}
	\item \label{ex:zeroRing} Prove that if $A$ is a ring such that $1 = 0$, then $A$ is the 0 ring.
	\item \label{ex:prop11} Prove Proposition 1.1. in [AM].
	\item Use the relation between ideals in $\ZZ$ and $\ZZ/(n) = \ZZ_n$ to show that the number of ideals in $\mathbb Z_n$ equals the number of positive integers dividing $n$.
	\item Let $k$ be a field, and prove that the ideal
	\[
	(x,y) = \{g_1x + g_2y \mid g_1, g_2 \in k[x,y]\} \subseteq k[x,y]
	\]
	is not a principal ideal.
	\item Let $\mathfrak a \subset \mathbb Z[x]$ be the set of polynomials such that $f \in \mathfrak a$ if and only if $f(0)$ is even.
	Show that $\mathfrak a$ is an ideal, and that it is not a principal ideal.
	\item Prove that a ring $A$ is an integral domain if and only if $A[x]$ is an integral domain.
	\item Convince yourself that for any ring $A$, we have an isomorphism $(A[x])[y] \cong A[x,y]$
	\item Let $A$ be a ring.
	Prove that there exists some field $k$ such that there is a surjective homomorphism $\phi \colon A \to k$.
	\item Let $A$ be a ring.
	Prove that there exists a unique homomorphism $\phi \colon \mathbb Z \to A$.
	\item Let $n \ge 1$, and let $\ZZ^n$ be the ring of $n$-tuples
	\[
	(a_1,\ldots, a_n) \ \ \ \ \ a_i \in \ZZ,
	\]
	with addition and multiplication defined componentwise.
	Prove that for any ring $A$ there are exactly $n$ homomorphisms $\phi \colon \ZZ^n \to A$.
	\item Let $A$ be a ring, and let $a \in A$.
	Prove that there exists a unique homomorphism $\phi \colon \mathbb Z[x] \to A$ such that $\phi(x) = a$.
	Use this to describe the set of all homomorphisms $\phi \colon \ZZ[x] \to A$.
	\item Let $C$ be the ring of continuous functions $f \colon \RR \to \RR$.
	Prove that for every $x \in \RR$, the set
	\[
	\mathfrak m_x = \{f \mid f(x) = 0\} \subset C
	\]
	is a maximal ideal.
	
	Let $\mathfrak a$ be the set of functions $f$ such that $f(x) = 0$ when $x$ is sufficiently large (i.e.~there exists an $y \in \RR$ such that $x \ge y \Rightarrow f(x) = 0)$).
	Prove that $\mathfrak a$ is an ideal.
	Use this to prove that $C$ has a maximal ideal not equal to $\mathfrak m_x$ for any $x \in \RR$.
\end{enumerate}
From Atiyah--MacDonald chapter 1: 7, 8, 9, 10, 12, 15, 16.

From Ellingsrud: 1.1, 1.10, 2.15, 2.20

\newpage

\section*{Suggested problems week 2}
[AM]: 1.1

[E]: 2.2, 2.5, 2.11, 2.14, 2.24, 2.26, 2.28, 2.50 (Here ``comaximal'' is a synonym for ``coprime'')

(*) means I suspect a problem is difficult (and you're not likely to miss anything important by skipping it).
\begin{enumerate}
	\item Let $A$ be a ring, let $a_1,\ldots, a_n \in A$ and $b_1,\ldots, b_m \in A$.
	Prove that
	\[
	(a_1,\ldots, a_n)(b_1,\ldots, b_m) = (a_1b_1,a_1b_2, \ldots, a_nb_m),
	\]
	where the sequence in the rightmost brackets contains every product $a_ib_j$ with $1 \le i \le n$ and $1 \le j \le m$.
	\item Let $A$ be a ring, and let $a \in A$. Prove that $A[x]/(x-a) \cong A$.
	\item Let $A$ be a ring, let $a,b \in A$. Prove that $A[x]/(x-a, b) \cong A/(b)$.
	\item Let $A$ be a ring and let $\fa, \fb$ be ideals.
	Prove that $\fa, \fb$ are coprime if and only if there is no maximal ideal $\mathfrak m$ with $\mathfrak a, \mathfrak b \subseteq \mathfrak m$.
	\item Let $k$ be a field, and let $f, g \in k[x]$.
	Prove that if $(f)$ and $(g)$ are coprime if and only if $f$ and $g$ have no common irreducible factor.\footnote{Recall that in $k[x]$ every ideal is principal, every polynomial can be factored into irreducible polynomials, and $h \in k[x] \setminus \{0\}$ is irreducible if and only if $(h)$ is prime.}
	\item Let $k$ be a field, and $0 \not= f \in k[x]$.
	Prove that $k[x]/(f)$ is isomorphic to a direct product of rings of the form $k[x]/(g^n)$, where $g$ is irreducible.
	\item Let $(a_1,b_1) \not= (a_2,b_2) \in \CC^2$.
	Prove that the ideals $(x-a_1,y-b_1)$ and $(x-a_2,y-b_2)$ in $\CC[x,y]$ are coprime.
	\item Show that the ideals $(x-2)$ and $(2x^2 - 2)$ in $\ZZ[x]$ are not coprime.
	\item Let $a_1,\ldots, a_n, b \in A$, and let $1 \le i, j \le n$ with $i \not= j$. Prove that
	\[
	(a_1,\ldots, a_n) = (a_1,\ldots, a_{j-1}, a_j + ba_i, a_{j+1},\ldots,a_n)
	\]
	\item Let $a_1,\ldots, a_n \in A$, let $1 \le i \le n$, and let $c \in A$ be a unit.
	Prove that
	\[
	(a_1,\ldots, a_n) = (a_1,\ldots, a_{i-1}, ca_i, a_{i+1},\ldots, a_n).
	\]
	\item (*) Let $a_1,\ldots, a_n \in A$, and let $(b_{ij})_{1\le i, j \le n}$ be an invertible matrix with coefficients in $A$.\footnote{This means that there exist $(c_{ij})_{1 \le i,j \le n}$ with $c_{ij} \in A$ such that
		\[
		\sum_{k=1}^n b_{ik}c_{kj} = \sum_{k=1}^n c_{ik}b_{kj} = \begin{cases} 1 \text{ if }i = j \\
			0 \text{ if }i \not= j
		\end{cases}.
		\]
	}
	Prove that
	\[
	(a_1,\ldots, a_n) = \left(\sum_{j=1}^nb_{1j}a_j, \sum_{j=1}^nb_{2j}a_j, \ldots, \sum_{j=1}^nb_{nj}a_j\right).
	\]
	\item (*) Let $A = \ZZ[x]/(x^2 + 1)$, and let $p \in \ZZ$ be a prime.
	Let $\phi \colon \ZZ \to A$ be the natural map, and show that $\phi((p))A \subseteq A$ is a prime ideal if and only if $-1$ is a quadratic residue modulo $p$, that is if and only if the equation $x^2 + 1$ admits solutions in $\ZZ/(p)$.
	\textit{Hint:} Show that $A/\phi((p))A \cong \ZZ/(p)[x]/(x^2 + 1)$ and consider what it means for this ring to be an integral domain.
	\item Let $\fa, \mathfrak p \subseteq A$ be ideals with $\mathfrak p$ prime such that $\mathfrak a$ is not contained in $\mathfrak p$.
	Prove that $(\mathfrak p :\fa) = \mathfrak p$.
	\item Let $p_1,\dots, p_k$ be distinct prime numbers, let $m = \prod_{i=1}^k p_i^{e_i}$ and $n = \prod_{i=1}^k p_i^{f_i}$ with $e_i, f_i \ge 0$.
	Prove that $((m) : (n)) = (r)$ with $r = \prod_{i=1}^k p_i^{\max(e_i-f_i,0)}$.
	\item(*) \label{ex:nilpotentAndPrime} This exercise proves the implication ``$f \in A$ nilpotent'' $\Rightarrow$ $f$ is contained in every prime ideal of $A$.
	
	We will assume that $f$ is not nilpotent, and construct a prime ideal not containing $f$.
	Let $N = \{f^n \mid n \ge 0\}$, and let $S$ be the set of ideals $\mathfrak a$ of $A$ such that $\mathfrak a \cap N = \varnothing$.
	\begin{enumerate}
		\item Prove that $S$ is non-empty.
		\item Use a Zorn's lemma argument to prove that $S$ has a maximal element under partial ordering.
		\item Let $\mathfrak p \in S$ be a maximal element, and let $g \not \in \mathfrak p$.
		Prove that we can find $n \ge 0, x \in A, y \in \mathfrak p$ such that
		\[
		f^n = y + xg.
		\]
		\item Prove that $\mathfrak p$ is a prime ideal not containing $f$.
	\end{enumerate}
\end{enumerate}
\end{document}